Econometric optimization engine

ABSTRACT

A preferred set of prices for a plurality of products, generally demand coefficients is provided. Cost data including activity-based costs is also received. A preferred set of prices that will provided a local optimum for the preferred set of prices which maximizes profit is then determined. In one embodiment, a sales model and a cost model are created. A plurality of rules is specified. The preferred set of prices is then determined based on the sales model, the cost model, and the plurality of rules.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to co-pending and concurrently filedapplication Ser. No. 09/742,472, filed Dec. 20, 2000, entitled “ImputedVariable Generator”, by Suzanne Valentine, Krishna Venkatraman, PhilDelurgio, Michael Neal, and Hau Lee, which is incorporated by referenceherein for all purposes.

This application is related to co-pending and concurrently filedapplication Ser. No. 09/741,956, filed Dec. 20, 2000, entitled“Econometric Engine”, by Hau Lee, Suzanne Valentine, Michael Neal,Krishna Venkatraman, and Phil Delurgio, which is incorporated byreference herein for all purposes.

This application is related to co-pending and concurrently filedapplication Ser. No. 09/741,957, filed Dec. 20, 2000, entitled“Financial Model Engine”, by Phil Delurgio, Suzanne Valentine, MichaelNeal, Krishna Venkatraman, and Hau Lee, which is incorporated byreference herein for all purposes.

This application is related to co-pending and concurrently filedapplication Ser. No. 09/741,958, filed Dec. 20, 2000, entitled “PriceOptimization System”, by Michael Neal, Krishna Venkatraman, SuzanneValentine, Phil Delurgio, and Hau Lee, which is incorporated byreference herein for all purposes.

BACKGROUND OF THE INVENTION

The present invention relates to providing a price optimization system.

In businesses, prices of various products must be set. Such prices maybe set with the goal of maximizing profit or demand or for a variety ofother objectives. Profit is the difference between total revenue andcosts. Total sales revenue is a function of demand and price, wheredemand is a function of price. Demand may also depend on the day of theweek, the time of the year, the price of related products, location of astore, and various other factors. As a result, the function forforecasting demand may be very complex. Costs may be fixed or variableand may be dependent on demand. As a result, the function forforecasting costs may be very complex. For a chain of stores with tensof thousands of different products, forecasting costs and determining afunction for forecasting demand are difficult. The enormous amounts ofdata that must be processed for such determinations are too cumbersomeeven when done by computer. Further, the methodologies used to forecastdemand and cost require the utilization of non-obvious, highlysophisticated statistical processes.

It is desirable to provide an efficient process and methodology fordetermining the prices of individual products such that profit (orwhatever alternative objective) is optimized.

SUMMARY OF THE INVENTION

To achieve the foregoing and other objects and in accordance with thepurpose of the present invention for determining a preferred set ofprices for a plurality of products, generally demand coefficients. Costdata is also received. A preferred set of prices that will provided alocal optimum is then determined.

In addition the present invention provides a method for determining apreferred set of prices for a plurality of products. A sales model iscreated. A cost model is created. A plurality of rules is specified. Thepreferred set of prices is then determined based on the sales model, thecost model, and the plurality of rules.

In addition, the present invention provides an optimization engine,useful in association with an econometric engine for creating a salesmodel and a financial model engine for creating a cost model. A ruletool, which stores a plurality of rule parameters is combined with aprice calculator connected to the rule tool, the financial model engine,and the econometric engine, where the price calculator determines apreferred set of prices based on rule parameters, the sales model, andthe cost model.

These and other features of the present invention will be described inmore detail below in the detailed description of the invention and inconjunction with the following figures.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not by wayof limitation, in the figures of the accompanying drawings and in whichlike reference numerals refer to similar elements and in which:

FIG. 1 is a high level schematic view of an embodiment of the invention.

FIG. 2 is high level flow chart of the invention.

FIG. 3 is a more detailed schematic view of the econometric engine.

FIG. 4 is a more detailed schematic view of the optimization engine andsupport tool.

FIG. 5 is a block diagram to illustrate some of the transaction coststhat occur in retail businesses of a chain of stores.

FIG. 6 is a flow chart of the preferred embodiment of the invention forproviding an initial feasible solution.

FIGS. 7A and 7B illustrate a computer system, which forms part of anetwork and is suitable for implementing embodiments of the presentinvention.

FIG. 8 is a schematic illustration of an embodiment of the inventionthat functions over a network.

FIG. 9A is a graph of original profit from actual sales of the storeusing actual prices and optimal profit from optimized sales resultingfrom the calculated optimized prices bounded by its probability.

FIG. 9B is a graph of percentage increase in profit and the probabilityof obtaining at least that percentage increase in profit.

FIG. 10 is a flow chart depicting a process flow by which raweconometric data can be input, subject to “cleansing”, and used tocreate an initial dataset which can then be used to generate imputedeconometric variables in accordance with a preferred embodiment of thepresent invention.

FIG. 11 is a flow chart depicting a process flow depicting a process bywhich partially cleansed econometric data is subject to further errordetection and correction in accordance with a preferred embodiment ofthe present invention.

FIG. 12A is a flow chart depicting a process flow by which an imputedbase price variable can be generated in accordance with one embodimentof the present invention.

FIG. 12B is a price time diagram which illustrates an aspect ofgenerating an imputed base price variable in accordance with oneembodiment of the present invention.

FIG. 12C is a price time diagram which illustrates an aspect ofgenerating an imputed base price step function in accordance with oneembodiment of the present invention.

FIG. 12D is a diagram which illustrates an error correction aspect ofbase price imputation in accordance with one embodiment of the presentinvention

FIG. 13 is a flow chart depicting a process flow by which an imputedrelative price variable can be generated in accordance with oneembodiment of the present invention.

FIG. 14A is a flow chart depicting a process flow by which an imputedbase unit sales volume variable can be generated in accordance with oneembodiment of the present invention.

FIG. 14B is a diagram used to illustrate the comparative effects ofsales volume increase and price discounts.

FIG. 15A is a flow chart depicting a process flow by which supplementaryerror detection and correction in accordance with an embodiment of thepresent invention.

FIG. 15B is a diagram used to illustrate the comparative effects ofsales volume increase and price discounts.

FIG. 16 is a flow chart depicting a process flow by which an imputedstockpiling variable can be generated in accordance with an embodimentof the present invention.

FIG. 17 is a flow chart depicting a process flow by which an imputedday-of-week variable can be generated in accordance with an embodimentof the present invention.

FIG. 18 is a flow chart depicting a process flow by which an imputedseasonality variable can be generated in accordance with an embodimentof the present invention.

FIG. 19A is a flow chart depicting a process flow by which an imputedpromotional effects variable can be generated in accordance with anembodiment of the present invention.

FIG. 19B is a diagram depicting the modeling effects of a promotionaleffects variable in accordance with an embodiment of the presentinvention.

FIG. 20 is a flow chart depicting a process flow by which an imputedcross-elasticity variable can be generated in accordance with apreferred embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention will now be described in detail with reference toa few preferred embodiments thereof as illustrated in the accompanyingdrawings. In the following description, numerous specific details areset forth in order to provide a thorough understanding of the presentinvention. It will be apparent, however, to one skilled in the art, thatthe present invention may be practiced without some or all of thesespecific details. In other instances, well known process steps and/orstructures have not been described in detail in order to notunnecessarily obscure the present invention.

I. OVERALL SYSTEM

To facilitate discussion, FIG. 1 is a schematic view of a priceoptimizing system 100. The price optimizing system 100 comprises aneconometric engine 104, a financial model engine 108, an optimizationengine 112, and a support tool 116. The econometric engine 104 isconnected to the optimization engine 112, so that the output of theeconometric engine 104 is an input of the optimization engine 112. Thefinancial model engine 108 is connected to the optimization engine 112,so that the output of the financial model engine 108 is an input of theoptimization engine 112. The optimization engine 112 is connected to thesupport tool 116 so that output of the optimization engine 112 isprovided as input to the support tool 116 and output from the supporttool 116 may be provided as input to the optimization engine 112. Theeconometric engine 104 may also exchange data with the financial modelengine 108.

FIG. 2 is a high level flow chart of a process that utilizes the priceoptimizing system 100. The operation of the optimizing system 100 willbe discussed in general here and in more detail further below. Data 120is provided from the stores 124 to the econometric engine 104 (step204). Generally, the data 120 provided to the econometric engine 104 maybe point-of-sale information, product information, and storeinformation. The econometric engine 104 processes the data 120 toprovide demand coefficients 128 (step 208) for a set of algebraicequations that may be used to estimate demand (volume sold) givencertain marketing conditions (i.e. a particular store in the chain),including a price point. The demand coefficients 128 are provided to theoptimization engine 112. Additional processed data from the econometricengine 104 may also be provided to the optimization engine 112. Thefinancial model engine 108 may receive data 132 from the stores 124(step 216) and processed data from the econometric engine 104. The data132 received from the stores is generally cost related data, such asaverage store labor rates, average distribution center labor rates, costof capital, the average time it takes a cashier to scan an item (orunit) of product, how long it takes to stock a received unit of productand fixed cost data. The financial model engine 108 may process the datato provide a variable cost and fixed cost for each unit of product in astore. The processing by the econometric engine 104 and the processingby the financial model engine 108 may be done in parallel. Cost data 136is provided from the financial model engine 108 to the optimizationengine 112 (step 224). The optimization engine 112 utilizes the demandcoefficients 128 to create a demand equation. The optimization engine isable to forecast demand and cost for a set of prices to calculate netprofit. The stores 124 may use the support tool 116 to provideoptimization rules to the optimization engine 112 (step 228). Theoptimization engine 112 may use the demand equation, the variable andfixed costs, and the rules to compute an optimal set of prices that meetthe rules (step 232). For example, if a rule specifies the maximizationof profit, the optimization engine would find a set of prices that causethe largest difference between the total sales and the total cost of allproducts being measured. If a rule providing a promotion of one of theproducts by specifying a discounted price is provided, the optimizationengine may provide a set of prices that allow for the promotion of theone product and the maximization of profit under that condition. In thespecification and claims the phrases “optimal set of prices” or“preferred set of prices” are defined as a set of computed prices for aset of products where the prices meet all of the rules. The rulesnormally include an optimization, such as optimizing profit oroptimizing volume of sales of a product and constraints such as a limitin the variation of prices. The optimal (or preferred) set of prices isdefined as prices that define a local optimum of an econometric modelwhich lies within constraints specified by the rules. When profit ismaximized, it may be maximized for a sum of all measured products. Sucha maximization, may not maximize profit for each individual product, butmay instead have an ultimate objective of maximizing total profit. Theoptimal (preferred) set of prices may be sent from the optimizationengine 112 to the support tool 116 so that the stores 124 may use theuser interface of the support tool 116 to obtain the optimal set ofprices. Other methods may be used to provide the optimal set of pricesto the stores 124. The price of the products in the stores 124 are setto the optimal set of prices (step 236), so that a maximization ofprofit or another objective is achieved.

Each component of the price optimizing system 100 will be discussedseparately in more detail below.

II. ECONOMETRIC ENGINE

FIG. 3 is a more detailed view of the econometric engine 104. Theeconometric engine comprises an imputed variable generator 304 and acoefficient estimator 308. The data 120 from the stores 124 is providedto the imputed variable generator 304. The data 120 may be raw datagenerated from cash register data, which may be generated by scannersused at the cash registers.

A. Imputed Variable Generator

The present invention provides methods, media, and systems forgenerating a plurality of imputed econometric variables. Such variablesare useful in that they aid businesses in determining the effectivenessof a variety of sales strategies. In particular, such variables can beused to gauge the effects of various pricing or sales volume strategies.

FIG. 10 illustrates a flowchart 1000 which describes steps of a methodembodiment for data cleansing imputed econometric variable generation inaccordance with the principles of the present invention. The process,generally described in FIG. 10, begins by initial dataset creation anddata cleaning (Steps 1011-1031). This data set information is then usedto generate imputed econometric variables (Step 1033) which can beoutput to and for other applications (Step 1035).

1. Initial Dataset Creation and Cleaning

The process of dataset creation and cleaning (that is to say the processof identifying incompatible data records and resolving the dataincompatibility, also referred to herein as “error detection andcorrection”) begins by inputting raw econometric data (Step 1011). Theraw econometric data is then subject to formatting and classifying byUPC designation (Step 1013). After formatting, the data is subject aninitial error detection and correction step (Step 1015). Once theeconometric data has been corrected, the store information comprisingpart of the raw econometric data is used in defining a store data sethierarchy (Step 1017). This is followed by a second error detecting andcorrecting step (Step 1019). This is followed by defining a group ofproducts which will comprise a demand group (i.e., a group of highlysubstitutable products) and be used for generating attribute information(Step 1021). Based on the defined demand group, the attributeinformation is updated (Step 1023). The data is equivalized and thedemand group is further classified in accordance with size parameters(Step 1025). The demand group information is subjected to a third errordetection and correction step (Step 1027). The demand group informationis then manipulated to facilitate decreased process time (Step 1029).The data is then subjected to a fourth error detection and correctionstep (Step 1031), which generates an initial cleansed dataset. Usingthis initial cleansed dataset, imputed econometric variables aregenerated (Step 1033). Optionally, these imputed econometric variablesmay be output to other systems for further processing and analysis (Step1035).

The process begins by inputting raw econometric data (Step 1011). Theraw econometric data is provided by a client. The raw econometric dataincludes a variety of product information. The raw econometric data mustspecify the store from which the data is collected, the time period overwhich the data is collected and include a UPC (Universal Product Code)for the product, and provide a UPC description of the product. Also, theraw econometric data must include product cost (e.g., the wholesale costto the store), number of units sold, and either unit revenue or unitprice. Ordinarily, the UPC description also identifies the productbrand, UOM (Unit of Measure), and product size. Such information can bevery detailed or less so. For example, brand can simply be Coca-Cola®,or more detailed e.g., Cherry Coca-Cola®. A UOM is, for example, ounces(oz.), pound (lb.), liter (ltr), or count (CT). Size reflects the numberof UOM's e.g., eight (8) oz or two (2) ltr. Also, the general categoryof product or department identification is input. A category is definedas a set of substitutable or complementary products, for example,“Italian Foods”. Such categorization can be proscribed by the client, ordefined by generally accepted product categories. Additionally, suchcategorization can be accomplished using look-up tables or computergenerated product categories.

Also, a more complete product descriptor is generated using the productinformation described above and, for example, a UPC description of theproduct and/or a product description found in some other look-up table(Step 1013). This information is incorporated into a product format.This product format provides a more complete picture of the product, butthis information is stored in a separate database which is notnecessarily processed using the invention. This information provides adetailed description of the product which can be called up as needed.

The data is then subjected to a first error detection and correctionprocess (Step 1015). Typically, this step includes the removal of allduplicate records and the removal of all records having no match in theclient supplied data (typically scanner data). An example of recordshaving no match are records that appear for products that the clientdoes not carry or stock in its stores. These records are detected anddeleted.

Data subsets concerning store hierarchy are defined (Step 1017). Thismeans stores are identified and categorized into various useful subsets.Typical subsets include (among other categorizations) stores segregatedby, for example, zip codes, cities, states, specific geographicalregions, rural environs, urban environs, associations with other stores(e.g., is this store part of a mall) or categorized by specific stores.A wide variety of other subsets may also be used. These subsets can beused to provide information concerning, among other things, regional orlocation specific economic effects.

The data is then subjected to a second error detection and correctionprocess (Step 1019). This step cleans out certain obviously defectiverecords. Examples include, but are not limited to, records displayingnegative prices, negative sales volume, or negative cost. Recordsexhibiting unusual price information are also removed. Such unusualprices can be detected using cross store price comparisons (betweensimilarly situated stores), for example, an average price for a productin the stores of a particular geographic region can be determined byaveraging the prices for all such products sold in the subject stores.The standard deviation can also be calculated. Prices that lie atgreater than, for example, two (2) standard deviations from the meanprice will be treated as erroneous and such records will be deleted.These tools can be applied to a variety of product parameters (e.g.,price, cost, sales volume).

This is followed by defining groups of products and their attributes andexporting this information to a supplementary file (e.g., a textfile)(Step 1021). This product information can then be output into aseparate process which can be used to define demand groups or productattributes. For example, this supplemental file can be input into aspreadsheet program (e.g., Excel®) which can use the product informationto define “demand groups” (i.e. groups of highly substitutableproducts). Also, further product attribute information can be acquiredand added to the supplementary file. Such attributes can comprise, forexample, branding information, manufacturer, size, flavor or form (e.g.,cherry soda) just to name a few. Such information can be gleaned frommultiple sources e.g., UPC product catalogues, the client, productlook-up tables, or other sources. The advantage of such supplementaryfiles is that they maintain complete product information (includinginformation not required by the processes of the present invention)which can be accessed when needed. In addition, updated demand group andattribute information can then be input as received (Step 1023). Bymaintaining a supplementary file containing large amounts of data, amore streamlined (abbreviated) dataset may be used in processing. Thiseffectively speeds up processing time by deleting non-criticalinformation from the dataset.

The data is further processed by defining an “equivalizing factor” forthe products of each demand group in accordance with size and UOMparameters (Step 1025). This equivalizing factor can be provided by theclient or imputed. An example of determining an imputed equivalizingfactor follows. Product size and UOM information are obtained, forexample, from the product description information. Typical examples ofsuch size and UOM information is, 20 oz. (ounce), 6 CT (count), or 1 ltr(liter). A further advantageous aspect of the present invention is that,even if such size or UOM information is incomplete or not provided, itcan also be imputed. An equivalizing factor can be imputed by using, forexample, the median size for each UOM. Alternatively, some commonly usedarbitrary value can be assigned. Once this information is gathered, allproduct prices and volume can be “equivalized”. In one example, a demandgroup (a group of highly substitutable products) is chosen having, forexample, “soft drinks” as its subject category. And by further example,the soft drink product comes in 8, 12, 16, 24, 32, and 64 ounce sizes.The median size (or for that matter, any arbitrarily determined size)can then be used as the base size to which all other sizes are to beequivalized. For example, using the 8, 12, 16, 24, 32, and 64-ouncesizes discussed above, an arbitrary base size can be determined as, forexample, 24 ounces. Then the 24-ounce size is determined as theequivalizing factor. Some of the uses of the equivalizing factors aredetailed in the discussions below. Chiefly, the purpose of determiningan equivalizing factor is to facilitate comparisons between differentsize products in a demand group. For example, if 16 is determined as theequivalizing factor for the above group of soft drinks, then an 8 oz.soft drink is equivalized to one half of a 16 oz. unit. In a relatedvein, a 32 oz. soft drink is equivalized to two (2) 16 oz. units.

Additionally, size information can be used to define further productattributes. For example, if the size is in the bottom tertile of sizesfor that product, it will be classified as “Small” size.Correspondingly, if the size is in the middle tertile of sizes for thatproduct, it will be classified as “Medium” size, and if the size is inthe top tertile of sizes for that product, it will be classified as“Large” size. Such categorization can define product attributes such assmall (8- and 12-ounce sizes), medium (16- and 24-ounce sizes), andlarge (32- and 64-ounce sizes) sizes.

The data is then subjected to a third error detection and correctionprocess, which detects the effects of closed stores and certain othererroneous records (Step 1027). Keeping in mind that one advantage of thepresent invention is that very little client input is required toachieve accurate results, the inventors contemplate error correctionwithout further input (or very little input) from the client. In accordwith the principles of the invention, stores that demonstrate no productmovement (product sales equal to zero) over a predetermined time periodare treated as closed. Those stores and their records are dropped fromthe process. In a preferred embodiment, the predetermined time period isthree (3) months.

With continued reference to FIG. 10, Step 1027, the third errordetection and correction also includes analysis tools for detecting thepresence of erroneous duplicate records. The data is examined, inparticular checking records for, date, product type, store at which theproduct was sold (or just “store”), price, units (which refers variouslyto units sold or unit sales volume), and causal variables. Causalvariables are those factors which influence sales volume (a variablewhich can cause an increase in product sales e.g., coupons, salespromotion ads, sale prices, sale price on some complementary product,enhanced sales displays, more advantageous sales location within astore, etc.). Analysis is performed to remove the discrepant recordssuch that only one of the records is kept as part of the analyzed dataand that causal information for a particular time period is recorded.

Using the following illustrative table:

Record Causal Number Date Store Product Units Price Variable 1 12/5  Y D10 1.99 1 2 12/5  Y D 10 1.99 1 3 12/12 Y D 10 1.99 1 4 12/12 Y D 151.89 2 5 12/19 Y D 12 1.99 1 6 12/26 Y D  9 1.99 1For example, using record #1, the date of record is 12/5, the store isstore “Y”, the product is product type “D”, units sold for that date are10 at a price of 1.99. The causal variable is usually abbreviated with acode symbol (e.g., numbers). Here, “1” is a symbol for no causalvariable, i.e., normal sales conditions. Whereas, examining, forexample, record #3 includes a causal variable (code symbol 2) which, forexample, will represent an advertisement concerning product “D”.

Discrepant records are identified and corrected. For example, if tworecords have the same exact values (such as record #1 and record #2), itis assumed that one such record is an erroneous duplicate and only onerecord is kept as part of the analyzed dataset, for example, only record#1 is retained.

If two records with the same date, product id, and store id havemultiple records with different casuals, they are combined into a singlerecord, with the two prices maintained in separate dataset variables,units summed across the two records, and the causal variablesrepresenting something other than a normal state being represented bynew dataset variables.

The following table shows an updated version of the above table. Record2 was deleted because it is identical to Record 1. Records 3 and 4 werecombined into a single record (i.e., combined into a single Record 3)with new causal variables defined for Advertisement and AdvertisementPrice. Records 5 and 6 did not change because there was no duplicateinformation.

Record Pro- Regular Advertise- Advertise- Number Date Store duct UnitsPrice ment ment Price 1 12/5  Y D 25 1.99 No — 3 12/12 Y D 25 1.99 Yes1.89 5 12/19 Y D 12 1.99 No — 6 12/26 Y D  9 1.99 No —

A further correction can be made for records having the same date andcausal value but have differing prices or differing number of unitssold. First, a data discrepancy must be detected. For example, if arecord on a specific date in the same store for the same product andcausal state has two different values for units, this is a discrepancy.Correction can be accomplished by, first calculating the average numberof units sold over all dates in the modeled time interval. Thediscrepant records are compared with the average value. The recordhaving the unit value closest to the calculated average units is keptand the other is discarded. The same general process can be followed forrecords having discrepancies as to price (i.e., the record having theprice closest to average price is kept). If both price and units aredetermined to have a discrepancy, the record having the price and unitvalues closest to the average price and average units is kept.

After all the duplicate records eliminated, the data is reconstructed.The data can be reviewed again to insure all duplicates are removed.Optionally, an output file including all discrepancies can be produced.In the event that it becomes necessary, this output file can be used asa follow-up record for consulting with the client to confirm theaccuracy of the error detection and correction process.

Additionally, reduced processing times may be achieved by reformattingthe data (Step 1029). For example, groups of related low sales volumeproducts (frequently high priced items) can optionally be aggregated asa single product and processed together. Additionally, the data may besplit into conveniently sized data subsets defined by a store or groupsof stores which are then processed together to shorten the processingtimes. For example, all stores in the state of California can beprocessed together, then all the stores in Texas, etc.

Next the process includes determining the nature of missing data recordsin a fourth error detection and correction step (Step 1031). The missingdata records are analyzed again before finally outputting a cleansedinitial dataset. For example, data collected over a modeled timeinterval is analyzed by introducing the data into a data grid dividedinto a set of time periods. The time periods can be preset, computerdetermined, or user defined. The time periods can include, but are notlimited to, months, weeks, days, or hours. One preferred embodiment usestime periods of one week. The data grid so constructed is analyzed. Forthe time periods (e.g., weeks) having no records a determination must bemade. Is the record missing because:

a. there were no sales that product during that week (time period);

b. the product was sold out and no stock was present in the store duringthat time period (this situation is also referred to herein as a“stock-out”);

c. the absence of data is due to a processing error.

FIG. 11 depicts a process flow embodiment for determining the nature ofmissing data records in a fourth error detection and correction step inaccordance with the principles of the present invention. The records arecompared to a grid of time periods (Step 1101). The grid is reviewed formissing records with respect to a particular store and product (Step1103). These missing records are then marked with a placeholder (Step1105). Missing records at the “edges” of the dataset do notsignificantly affect the dataset and are deleted (Step 1107). Recordsfor discontinued products or products recently introduced are droppedfor those time periods where the product was not carried in the Store(Step 1109). The remaining dataset is processed to determine an averagevalue for units (sold) and a STD for units (Step 1111). Each missingrecord is compared to the average units (Step 1113) and based on thiscomparison, a correction can be made (Step 1115).

Referring again to FIG. 11, in Step 1101, the data records are matchedwith a grid of time periods (shown here as weeks, but which can be anychosen time period). The grid can cover an entire modeled time interval,for example, as shown below, the six weeks January 7-February 14 (shownhere as weeks 1, 2, 3, 4, 5, and 6). Each product in each store (herestore “Z”) is gridded this way. For example:

Grid Date Store Product Units Price 1 1/7  Z Y 10 1.99 2 1/14 Z Y 122.19 3 4 1/28 Z Y  8 1.99 5 2/7  Z Y 10 1.99 6

Review of the grid (Step 1103) shows that records are “missing” fordates January 21 and February 14 (i.e., grid 3 and grid 6). Placeholdersare set in the records defined by grid 3 and grid 6 (Step 1105). Forexample, an easily detectable or arbitrarily large value can be put inthe price column of the grid, e.g. 999. Alternatively, a simple X can beplaced as a placeholder in the price column. In the present example,“X's” can be placed in the price columns of grid 3 and grid 6.

If the first or last grid in the dataset (here grid 1 or grid 6) has fewor no observations, those records are deleted from the dataset (Step1107). For purposes of the above analysis, a grid having “few”observations is defined as a grid having 50% fewer observations than isnormal for the grids in the middle of the dataset. Here, for example,the record for grid 6 (the last week) is deleted because no records arepresent for that week. Also, using client-supplied stocking information,products which have been discontinued during the modeled time intervaldo not have their grids filled out for the discontinued time period(Step 1109). Also, products which are introduced during the modeled timeinterval have their time grid filled out only for those time periodsoccurring after the product introduction date. Thus, certain dataaberrations are removed from the modeled dataset, permitting moreaccurate modeling.

The mean units (sold) and the STD for units are then calculated (Step1111). For example, in dataset depicted above, the mean is 10 units. Themissing record is then compared with the mean value (Step 1113). Here, amissing record (grid 3) is assigned an initial unit value=0. If thevalue of zero units lies within one (1) STD of the calculated mean, itis assumed that an actual value of zero units is feasible and thatrecord is treated as if the record is valid (unit volume=0). However, ifzero lies at greater than one STD from the mean, it is assumed that thevalue of zero units is due to a “stock-out”. In such case, it is assumedthat had product been present in the store an average number of unitswould have been sold. Therefore, the zero unit value for that record isreplaced by a unit value equal to the calculated mean unit value,thereby correcting for the “stock-out”. In this case, units for grid 3will be corrected to calculated mean units (i.e., 10).

The product histories of the dataset can also be examined. If thesubject product was introduced or discontinued as a salable item at thesubject store during the modeled time interval, the grid is not filledout (with either zero or average values) for those time periods wherethe product was not offered for sale in the subject store. In this waymissing records do not corrupt the dataset.

Further aspects of the fourth error detection and correction include adetection and elimination of outlying price data points (outliers). Asatisfactory way of accomplishing this begins with a calculation of themean price for each product within a given store, as determined over themodeled time interval. Once a mean price and STD are determined, allprice data for the modeled time interval is examined. If it isdetermined that a price record lies within three (3) STD from the meanprice it is deemed accurate and not an outlier. However, prices lyingoutside three (3) STD are treated as outliers. These outliers areassigned adjusted prices. The adjusted prices have the value of theimmediately preceding time period (e.g., the previous day's or week'sprice for that product within the store). This adjusted price data isagain checked for outliers (using the original mean and STD). Again,outliers are checked against the original mean and STD and again priceadjusted if necessary. This usually removes all the remaining outliers.However, the process may optionally continue, iteratively, until thereare no further outliers.

The net result of execution of the process Steps 1011-1031 disclosedhereinabove is the generation of a cleansed initial dataset which can beused for its own purpose or input into other econometric processes. Onesuch process is the generation of imputed econometric variables.

2. Generation of Imputed Econometric Variables

The foregoing steps (1011-1031) concern cleansing the raw econometricdata to create an error detected and error corrected (“cleansed”)initial dataset. The cleansed initial dataset created in the foregoingsteps can now be used to generate a variety of useful imputedeconometric variables (Step 1033). These imputed econometric variablesare useful in their own right and may also be output for use in furtherprocessing (Step 1035). One particularly useful application of theimputed econometric variables is that they can be input into anoptimization engine which collects data input from a variety of sourcesand processes the data to provide very accurate economic modelinginformation.

a. Imputed Base Price

One imputed econometric variable that can be determined using theinitial dataset created in accordance with the forgoing, is an imputedbase price variable (or base price). FIG. 12A is a flowchart 1200outlining one embodiment for determining the imputed base pricevariable. The process begins by providing the process 1200 with a“cleansed” initial dataset (Step 1201), for example, the initial datasetcreated as described in Steps 1011-1031 of FIG. 10. The initial datasetis examined over a defined time window (Step 1203). Defining a timewindow (Step 1203) includes choosing an amount of time which frames aselected data point allowing one to look forward and backward in timefrom the selected data point which lies at the midpoint in the timewindow. This is done for each data point in the dataset, with the timewindow being defined for each selected data point. The time frame can beuser selected or computer selected. The time window includes T timeperiods and the time period for the selected data point. One preferredset of T time periods is eight (8) weeks. It is contemplated that timewindows of greater or lesser size can be selected. Referring to apreferred example, the selected (or current) data point is centered inthe time window having T/2 time periods before the selected data pointand T/2 time periods after the selected data point. In the presentexample, the time window includes the four weeks preceding the selecteddata point and the four weeks after the selected data point.

Referring to FIG. 12B, the selected data point “X” (shown as a singleweek) is framed by a time period of −T/2 (shown here as 4 weeks) beforethe data point “X” and a time period of +T/2 (shown here as 4 weeks)after the data point “X”. The time window comprising all the time (i.e.,−T/2, X, T/2) between points a and b.

Referring again to FIG. 12A, once the time window is defined, an“initial base price” is determined (Step 1205). This can be accomplishedby the following process. With reference to FIG. 12B, two price maximaare determined (M₁, M₂), one for each of the T/2 time periods before andafter the current data point. The lesser value of the two maxima (hereM₁) comprises the initial base price. The actual price (in selected datapoint “X”) is compared with this initial base price (here, M₁). Ifinitial base price is higher than the actual price (as shown in thepictured example), then the “initial base price” is reset to reflect theprice for the previous time period. In the pictured example, the lessermaxima M₁ is $1.00, the actual price during the data point “X” is lessthan $1.00 so the initial base price is reset to the price of theprevious time period “P” (here $1.00).

Alternatively, the initial base price can be determined using othermethods. For example, the average price of the product over the −T/2time period (4 weeks) preceding the data point X may be used as theinitial base price. Whatever method used, the initial base price isgenerated for each time period of the modeled time interval. One by one,each data point in the modeled time frame is examined and an initialbase price is determined for each time period (e.g., “X”) in the modeledtime interval.

The initial base price values generated above provide satisfactoryvalues for the imputed base price variable which may be output (Step1207) and used for most purposes. However, optional Steps 1209-1217describe an approach for generating a more refined imputed base pricevariable.

In generating a more refined imputed base price variable, the effect ofpromotional (or discount) pricing is addressed (Steps 1209-1217). Thismay be calculated by specifying a discount criteria (Step 1209);defining price steps (Step 1211); outputting an imputed base pricevariable and an imputed discount variable (Step 1213); analyzing thebase price distribution (Step 1215); and outputting a refined base pricevariable (Step 1217).

Data records are evaluated over a series of time periods (e.g., weeks)and evaluated. The point is to identify price records which arediscounted below a base price. By identifying these prices and notincluding them in a calculation of base price, the base pricecalculation will be more accurate. Therefore, a discount criterion isdefined and input as a variable (Step 1209). A preferred criterion is2%. Therefore, records having prices which are discounted 2% below thepreviously determined initial base price are treated as records having“promotional prices”. These records are temporarily deleted from thedataset. The remaining records, having zero or small discounts, aretreated as “non-promoted” records. So the price of each product for the“non-promoted” time periods (weeks) is averaged over all time periods(weeks) in the modeled time interval. The average non-promoted price isreferred to as a base price.

Further analysis is used to define base price “steps” (Step 1211). Thisprocess can be more readily illustrated with references to FIG. 12Cwhich shows a distribution of base price data points 1220, 1221, 1222and their relationship to a projected step function 1230, 1231, 1240,1241 plotted on a graph of price over time. Base price data points 1220,1221, 1222 are evaluated. Steps 1230, 1231 are roughly defined such thatthe base price data points 1220, 1221 lie within a small percent ofdistance from the step 1230, 1231 to which they are associated (e.g.,2%). This can be accomplished using, for example, a simple regressionanalysis such as is known to those having ordinary skill in the art. Bydefining the steps 1230, 1231, the average value for base price over thestep is determined. For example, price data points 1220 are averaged todetermine the base price of step 1230. Also, price data points 1221 areaveraged to determine the base price of step 1231. Thus, the average ofthe base prices in a step is treated as the refined base price for thatstep.

Further refining includes an analysis of the first step 1240. If thefirst step 1240 is short (along the time axis) and considerably lowerthan the next step 1230, it is assumed that the first step 1240 is basedon a discounted price point 1222. As such, the value of the next step1230 is treated as the base price for the time period of the first step1241 (represented by the dashed line).

At this point, absolute discount (ΔP) and base price (BP) are used tocalculate percent discount (ΔP/BP) for each store product time period.Percent discounts that are less than some value (e.g. 1%) are treated asbeing no discount and corrected to ΔP/BP=0. The above determined baseprice variable and percent discount variable are then output (Step1213).

This base price is subjected to further analysis for accuracy usingcross-store checking (Step 1215). This can be accomplished by analyzingthe base price data for each product within a given store. A curve isgenerated for each product. This curve defines the price distributionfor each product. The 80^(th) percentile for base price is thencalculated for the analyzed product (i.e., the base price point belowwhich 80% of the analyzed product (over the modeled time interval) ispriced). This is referred to as the “in store 80^(th) percentile” forthat product. A calculation is then made of the average 80^(th)percentile for price of the analyzed product across all stores (thecross-store 80^(th) percentile). Each store's prices are then mergedwith each other store to calculate the average 80^(th) percentile forbase price over all stores.

The stores are then analyzed product by product. If the base price for astore is greater than two (2) standard deviations from the cross-storeaverage 80^(th) percentile for base price and if the in-store 80^(th)store percentile is more than 50% different from the cross-store 80^(th)percentile, this store is flagged as an outlier for the analyzedproduct.

Store Product In Store 80^(th)% Cross-Store 80^(th)% Flagged Y A 1.991.99 No Y B 2.09 1.99 No Y C 0.29 1.99 Yes Y D 1.89 1.99 No

The outlier store's base price is adjusted for the analyzed product suchthat it lies only two (2) standard deviations away from the averagecross-store 80^(th) percentile for base price over all stores. This isillustrated in FIG. 12D. The average 80^(th) percentile price over allstores is shown as “Q”. If a flagged store has a base price for ananalyzed product beyond two (2) STD from the mean, as shown by datapoint 1250, that data point is corrected by moving the data point to the“edge” at two (2) STD (as shown by the arrow) from the mean. That point1251 is shown having a new base price of V.

Thus, the forgoing process illustrates an embodiment for determining animputed base price variable.

b. Imputed Relative Price Variable

Reference is now made to the flowchart 1300 of FIG. 13 which illustratesan embodiment for generating relative price variables in accordance withthe principles of the present invention. In the pictured embodiment, theprocess begins with a calculation of an “equivalent price” for eachproduct sold for each store (Step 1301). The following example will usesoda to illustrate an aspect of the present invention. An exampledataset is shown below:

Equivalent Actual Equivalent Equivalent Product Size Factor Price UnitsUnits Price A  8 16 1.00 500 250 2.00 B 16 16 2.00 300 300 2.00 C 32 163.00 100 200 1.50

Using this data, relative price may be calculated. As disclosed earlier,an equivalizing factor is defined. For this example, let theequivalizing factor be 16. Using the equivalizing factor, an equivalentprice can be calculated (Step 1301).

${{Equivalent}\mspace{14mu}{Price}} = {{Actual}\mspace{14mu}{{Price} \cdot \left( \frac{Equivalizingfactor}{size} \right)}}$Thus  for${A\text{:}\mspace{25mu}{Equivalent}\mspace{14mu}{Price}} = {{{\$ 1}{.00}\left( \frac{16}{8} \right)} = {{\$ 2}{.00}}}$${B\text{:}\mspace{31mu}{\$ 2}{.00}\left( \frac{16}{16} \right)} = {{\$ 2}{.00}}$${C\text{:}\mspace{31mu}{\$ 3}{.00}\left( \frac{16}{32} \right)} = {{\$ 1}{.50}}$the results of these calculations are shown in the “Equivalent Price”column of the table above.

Next equivalent units sold (“units”) can be calculated (Step 1303).

${{Equivalent}\mspace{14mu}{Units}} = {{units} \cdot \left( \frac{size}{{equivalizingfactor}\;} \right)}$Thus  for${A\text{:}\mspace{25mu}{Equivalent}\mspace{14mu}{units}} = {{500\left( \frac{8}{16} \right)} = 250}$${B\text{:}\mspace{25mu} 300 \times \left( \frac{16}{16} \right)} = 300$${C\text{:}\mspace{20mu} 100 \times \left( \frac{32}{16} \right)} = 200$

In a similar vein, equivalent base price and equivalent base units arecalculated (Step 1305) using the imputed values for base price (forexample, as determined in Steps 1201-1207) and for base units (alsoreferred to as base volume which is determined as disclosed below).

For each Store, each demand group, and each date, the total equivalentunits is determined (Step 1307). For example, using the dataset above(assuming that the data is from the same store), a total of 750 (i.e.,250+300+200) equivalent units were sold.

Defining A, B, and C as products in a demand group, the equivalentvalues for the demand group are depicted below:

Product Equivalent Units Equivalent Price A 250 $2.00 B 300 $2.00 C 200$1.50

A weighted calculation of relative equivalent price is then made (Step1309). For example, such relative price value is determined as follows:

Equivalent price is divided by a weighted denominator.

The weighted denominator is calculated by multiplying equivalent unitsfor each product times the equivalent units sold. For each product, onlythe values of other products are used in the calculation. This meansexcluding the product being analyzed. For example, if products A, B, andC are being analyzed in turn, when product A is analyzed the value for Ais excluded from the denominator. Using the above data, the relativeprice of A is determined as follows:

$\quad\begin{matrix}{{rel}_{A} = \frac{{equiv}.{priceofA}}{\left\lbrack \frac{{\left( {{equiv}.{unitsofB}} \right)\left( {{Equiv}.{priceofB}} \right)} + {\left( {{equiv}.{unitsofC}} \right)\left( {{equiv}.{priceofC}} \right)}}{{totalequivalentunits} - {equivalentunitsofA}} \right\rbrack}} \\{= \frac{2}{\left\lbrack \frac{{(300)(200)} + {(200)(1.50)}}{\left( {250 + 300 + 200} \right) - 250} \right\rbrack}} \\{= 1.111} \\{{rel}_{B} = \frac{2}{\left\lbrack \frac{{(250)(2.00)} + {(200)(1.50)}}{750 - 300} \right\rbrack}} \\{= 1.125} \\{{rel}_{C} = \frac{1.50}{\left\lbrack \frac{{(250)(2.00)} + {(300)(2.00)}}{750 - 200} \right\rbrack}} \\{= 0.75}\end{matrix}$

To insure that all members of a demand group are counted at least atsome minimal level, if equivalent units=0, a value of “1” is added toall units. In an example where equivalent units were A=0; B=5; C=11, theunits would be revalued as A=1; B=6; C=12, and the calculations asdisclosed above would be conducted. Also, where the number of productsin a demand group is equal to one, the weighted average equivalent pricefor the single product is equal to the equivalent price for thatproduct. If a value for equivalent price is missing, the equivalentprice for the previous record is used for equivalent price.

Also, a weighted average equivalent base price is calculated using themethod disclosed hereinabove. The only difference being that instead ofusing the actual equivalent price, the calculated base price values perequivalent are used (Step 1311). Using the previously disclosedtechniques, a moving average is generated for relative actual equivalentprice and relative equivalent base price (Step 1313).

This moving average is generally calculated by first defining a timeperiod window framing each analyzed date (e.g., four weeks, two weeksprior, two weeks after). This framing time period is specified as aninput. Second, for each date in the time period window, a weightedaverage of actual equivalent price and a weighted average of equivalentbase price are calculated. For time period windows where there areinsufficient days preceding the analyzed date (e.g., if the time windowrequires two week's worth of data but only one week is available),imputed values are provided for base price or actual price. Such imputedvalues are just the average value for base price or actual price,respectively. Third, once the time period window is defined,calculations are made defining average relative actual equivalent priceand average relative equivalent base price over the time period window,thereby defining a moving average for both relative actual equivalentprice and relative equivalent base price. This, is repeatedly done withthe window being moved incrementally through the dataset therebyobtaining a moving average.

Thus a variety of imputed relative price variables can be generated(e.g., relative equivalent price, relative equivalent base price, etc.).

C. Imputed Base Volume Variable

A flowchart 1400 shown in FIG. 14A illustrates one embodiment forgenerating an imputed base volume variable. Base volume refers to thevolume of product units sold in the absence of discount pricing or otherpromotional effects. Base volume is also referred to herein as simply“base units”. The determination of base volume begins by receiving thecleansed initial dataset information for each product and store (Step1401). The initial dataset information is processed to determine“non-promoted dates” (Step 1403). For example, using the percentdiscount (ΔP/BP) information generated above, product records having apercent price discount that is less than some predetermined discountlevel (e.g., 2%) are treated as non-promoted products for the timeperiods where the percent discount is less than the predetermineddiscount level (e.g., 2%). These records are used to generate a datasubset defining the dates where the products are not significantly pricediscounted i.e., “non-promoted dates”. This data subset is also referredto herein as the non-promoted data subset.

Using the non-promoted data subset, an average value for “units” and aSTD is calculated (i.e., an average value for product unit sales volumefor each product during the non-promoted dates is calculated) (Step1405). The average units are rounded up to the nearest integer value,this value shall be referred to as the “non-promoted average units”.

An initial value for base units (“initial base units”) is now determined(1407). This value is determined for all dates in the dataset, not justthe non-promoted dates. For those records having a percent pricediscount that is less than the predetermined discount level (e.g., 2%)the actual units sold are treated as “initial base units”. However,where such records (those the 2% or less discount) also have an actualvalue for units sold which is greater than 1.5 STD from the non-promotedaverage unit value (as calculated above), then the actual value forunits sold is not used. Instead, it is replaced with the non-promotedaverage unit value in calculating “initial base units”. For the otherrecords (those having a percent price discount that is equal to orgreater than the predetermined discount level (e.g., 2%)), thepreviously calculated non-promoted average unit value is used for“initial base units”.

This principle can be more readily understood with reference to FIG.14B. The price behavior 1450 can be compared with sales behavior 1460.Typically, when the price drops below a certain level, sales volumeincreases. This can be seen at time periods 1470, 1471. This can bereflective of, for example, a 2% or greater price discount. This is tobe expected, and as a result, these sales records should not affectcalculations of “base volume”. In such a case, the actual units sold(more than usual) are not included in a base volume determination.Rather, those records are replaced with the average volume value for thenon-promoted dates (the non-promoted average unit value, shown with thedotted lines 1480, 1481). However, where a sales volume increases duringa period of negligible discount (e.g., less than 2%), such as shown fortime period 1472, the actual units sold (actual sales volume) are usedin the calculation of base volume. However, if the records show a salesvolume increase 1472 which is too large (e.g., greater than 1.5 standarddeviations from the non-promoted average unit value), it is assumed thatsome other factor besides price is influencing unit volume and theactual unit value is not used for initial base units but is replaced bythe non-promoted average unit value.

A calculated base volume value is now determined (Step 1409). This isaccomplished by defining a time window. One preferred window is four (4)weeks, but the time window may be larger or smaller. For each store andproduct, the average value of “initial base units” is calculated foreach time window. This value is referred to as “average base units”.This value is calculated for a series of time windows to generate amoving average of “average base units”. This moving average of theaverage base units over the modeled time interval is defined as the“base volume variable”.

d. Supplementary Error Detection and Correction

Based on previously determined discount information, supplementary errordetection and correction may be used to correct price outliers. Aflowchart 1500 illustrated in FIG. 15A shows one embodiment foraccomplishing such supplementary error detection and correction. Suchcorrection begins by receiving the cleaned initial dataset informationfor each product and store (Step 1501). In addition the previouslycalculated discount information is also input, or alternatively, thediscount information (e.g., P/BP) can be calculated as needed. Theinitial dataset and discount information is processed to identifydiscounts higher than a preselected threshold (e.g., 60% discount) (Step1503). For those time periods (e.g., weeks) having price discountshigher than the preselected threshold (e.g., greater than 60%), acomparison of actual units sold to calculated base volume units (ascalculated above) is made (Step 1505).

The concepts are similar to that illustrated in FIG. 14B and may be moreeasily illustrated with reference to FIG. 15B. The principles of thisaspect of the present invention are directed toward finding unexplainedprice aberrations. For example, referring to FIG. 15B, price discountsare depicted at data points 1550, 1551, 1552, and 1553. Also,corresponding sales increases are depicted by at data points 1561, 1562,and 1563. The data point 1550 has a discount greater than the threshold1555 (e.g., 60%). So an analysis is made of data point 1550.

If the number of actual units sold (shown as 1560) lies within two (2)STD of the calculated base volume, then it is assumed that the actualprice 1550 is actually an erroneous record and the actual value 1560 isreplaced with the calculated base price.

However, if the number of actual units sold is greater than two (2) STDfrom the calculated base volume, it is assumed that the volume number iscorrect and the price is reset to reflect a discount of 60% and theprice is recalculated based on the 60% discount. In short, the discountis capped at the chosen value (here 60%). Once the data is corrected, itcan be output (step 1507).

e. Determining Imputed Variables which Correct for the Effect ofConsumer Stockpiling

With reference to FIG. 16, a flowchart 1600 illustrating a methodembodiment for generating stockpiling variables is depicted. Thepictured embodiment 1600 begins by defining the size of a “timebucket”(m), for example, the size (m) of the bucket can be measured indays (Step 1601). A preferred embodiment uses a bucket of one (1) weekor seven (7) days. Additionally, the number (τ) of time buckets to beused is also defined (Step 1603). The total amount of time “bucketed” (mX τ) is calculated (Step 1605).

“Lag” variables which define the number of product units sold (“units”)in the time leading up to the analyzed date are defined (Step 1607). Forexample:

Lag1(units)=number of product units sold in one (1) time period (e.g., aday or week) before the analyzed date;

Lag2(units)=number of product units sold in two (2) time periods (e.g.,a day or week) before the analyzed date;

-   -   .    -   .    -   .        Lagt(units)=number of product units sold in t time periods        (e.g., a day or week) before the analyzed date.

Then the total number of product units sold is calculated for eachdefined time bucket (Step 1609). For example:

Bucket1=sum of units sold during the previous m days;

Bucket2=sum of units sold from between the previous m+1 days to 2m days;

Bucket3=sum of units sold from between the previous 2 m+1 days to 3mdays;

-   -   .    -   .    -   .

Bucket(τ)=sum of units from between the previous (τ−1)m+1 days to (τ)mdays.

Correction can be made at the “front end” of the modeled time interval.For example, the data can be viewed as follows:

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Bucket — Week 1 Week 2Week 3 Week 4 Week 5 Week 6 1 Bucket — — Week 1 Week 2 Week 3 Week 4Week 5 2 Bucket — — — Week 1 Week 2 Week 3 Week 4 3 Bucket — — — — Week1 Week 2 Week 3 4

If working near the front end of a dataset, units from previous weekscannot always be defined and in their place an averaged value for bucketsum can be used (Step 1611). For example, referring to Bucket 1, thereis no Bucket 1 data for Week 1. As a result, the Bucket 1 data for weeks2-7 are averaged and that value is put into Week 1 of Bucket 1.Similarly, with reference to Bucket 2, Week 1 and Week 2 are missing avalue for Bucket 2, Weeks 1-3 are missing a value for Bucket 3, andWeeks 1-4 are missing a value for Bucket 4. The average values aregenerated for the missing values from weeks. For example, for Bucket 2,an average value for Weeks 3-7 is generated. This average value is usedto fill out the missing dates of Bucket 2 (Weeks 1-2). Similarly, forBucket 3, the average value for Weeks 4-7 are averaged and used to fillout the missing dates (Weeks 1-3). The same principle applies to Bucket4. These Buckets define variables which are used to model the impact ofpromotional activity in previous time periods. The Buckets are used asvariables in models which can be used to determine if there is arelationship between sales volume between a previous time as comparedwith a current time. The idea is to detect and integrate the effects ofconsumer stockpiling on into a predictive sales model.

f. Day of the Week Analysis

With reference to FIG. 17, a flowchart 1700 illustrating one embodimentfor determining a Day of the Week variable is shown. It is necessary tohave data on a daily basis for a determination of Day of the Weekeffects. In accordance with the principles of the present invention theembodiment begins by assigning the days of the week numerical values(Step 1701). A first date in the dataset is assigned. This can bearbitrarily assigned, but typically the first date for which data isavailable is selected as the “first date”. This date is assigned Day ofWeek “1”. The next six days are sequentially assigned Days of the Week2,3,4,5,6,7, respectively. This process continues with the nextconsecutive days data starting over again with Day of Week=“1”,continuing throughout all the days of the modeled time interval.

Once categorized by day of the week the product units (sold) are summedfor a specified dimension or set of dimensions. Dimension as used hereinmeans a specified input variable including, but not limited to, Product,Brand, Demand Group, Store, Region, Store Format, and other inputvariable which may yield useful information (Step 1703). For example, ifRegion is the specified dimension (e.g., all the stores in Los Angeles,Calif.), all of the unit volume for selected products in the Los Angelesstores is summed for each Day of Week (i.e., 1,2,3,4,5,6, and 7).

For each Day of Week and each dimension specified, the average units(sold) are determined (Step 1705). For each date, a “relative dailyvolume” variable is also determined (Step 1707). For example, relativedaily volume for a given Store is provided by (total Store dailyunits)/(average Store daily units). Such calculation can be accomplishedfor any input variable.

One numeric example can be shown as follows. A store sells 700 units ofX over a given modeled time interval. Average daily units=700/7=100. Ifsales for all of the Friday's of the modeled time interval amount to 150units, it can be shown that, for that Store, Friday's relative dailyvolume is 1.5, i.e., more than average. This information may provevaluable to a client merchant and can comprise an input variable forother econometric models.

g. Imputed Seasonality Variable Generation

Another useful imputed variable is an imputed seasonality variable fordetermining seasonal variations in sales volume. One preferred approachfor generating this variable is in accordance with the method describedby Robert Blattberg and Scott Neslin in their book “Sales Promotion:Concepts, Methods, and Strategies”, at pages 237-250 (Prentice Hall,N.J., 1990).

Referring to FIG. 18, a flowchart 1800 illustrating one embodiment inaccordance with the present invention for determining an imputedseasonality variable is shown. The process begins with categorizing thedata into weekly data records, if necessary (Step 1801). Zero values andmissing records are then compensated for (Step 1803). “Month” variablesare then defined (Step 1805). A logarithm of base units is then taken(Step 1807). Linear regressions are performed on each “Month” (Step1809). “Months” are averaged over a specified dimension (Step 1811).Indexes are averaged and converted back from log scale to original scale(Step 1813). The average of normalized estimates are calculated and usedas Seasonality index (Step 1815). Individual holidays are estimated andexported as imputed seasonality variables (Step 1817).

The embodiment begins by categorizing the data into weekly data records.Chiefly, this comprises aggregating daily data into weekly groups (Step1801). For missing sales records or records having zero volume values,insert average volume data (Step 1803).

A set of month variables is first defined (Step 1805). A series ofmodels of base units are constructed using each defined month variableas the predictor.

The process of defining month variables is as follows:

-   1) Define the month variable    -   a. Starting with Week 1, Day 1, assign a month number to each        week (Month1)    -   b. Assume 4 weeks per month    -   c. Depending on the time frame of the dataset, there may be 12        or 13 months defined-   2) Repeat definition of month variable three more times    -   a. Advance Week 1 to the second week in the dataset    -   b. Assign a month number to each week (Month2)    -   c. Advance Week 1 to the third week in the dataset    -   d. Assign a month number to each week (Month3)    -   e. Advance Week 1 to the fourth week in the dataset    -   f. Assign a month number to each week (Month4)

Week Month1 Month2 Month3 Month4 1 1 12  12  12  2 1 1 12  12  3 1 1 112  4 1 1 1 1 5 2 1 1 1 6 2 2 1 1 7 2 2 2 1 8 2 2 2 2 . . . . . . . . .. . . . . .

The values determined for base units are now processed. By taking thelog of base units the effect of extreme variations in base units can bereduced (Step 1807). A linear regression is run on the log of base unitsfor Month 1 (Step 1809). The regression models the log of base units asa function of Month 1 levels and Week number: (log(base units)=f(Month1,Week number)). The regression analysis is repeated using months Month2,Month3, and Month4 to determine, respectively log(base units)=f(Month2,Week number); log(base units)=f(Month3, Week number); and log(baseunits)=f(Month4, Week number).

-   3) The average value across the 12 (or 13) levels of the    Month1-Month4 estimates within the specified dimension (e.g. demand    group) is calculated.-   4) The estimates are indexed to the average estimate value and the    indexes are converted back to original scale:    -   a. Seasindx1=exp(estimate of Month1−avg. estimate of Month1)    -   b. Seasindx2=exp(estimate of Month2−avg. estimate of Month2)    -   c. Seasindx3=exp(estimate of Month3−avg. estimate of Month3)    -   d. Seasindx4=exp(estimate of Month4−avg. estimate of Month4)-   5) The average of the four normalized estimates is output as the    Final Seasonality index    -   a. Seasindx=Avg.(Seasindx1, Seasindx2, Seasindx3, Seasindx4)    -   b. The values for Seasindx will be centered around 1.0, and        typically range from 0.7 to 1.3.-   6) After estimating individual holidays, combine estimates with    index prior to tool export

h. Imputed Promotional Variable

Another useful variable is a variable which can predict promotionaleffects. FIG. 19A provides a flowchart illustrating an embodimentenabling the generation of imputed promotional variables in accordancewith the principles of the present invention. Such a variable can beimputed using actual pricing information, actual product unit salesdata, and calculated value for average base units (as calculated above).This leads to a calculation of an imputed promotional variable whichtakes into consideration the entire range of promotional effects.

FIG. 19B provides a useful pictorial illustration depicting arelationship between product price 1950, calculated average base units1951, and actual units sold 1952 and the results of a simple regressionmodel 1953 used to predict actual sales volume.

Referring back to FIG. 18A, the process begins by inputting the cleansedinitial dataset and the calculated average base units information (Step1901). A crude promotional variable is then determined (Step 1903). Sucha crude promotional variable can be defined using promotion flags. Thesepromotion flags may be set by an analysis of the unit sales for eachdate. If the actual unit sales (1952 of FIG. 19B) are greater than two(2) STD's from the average base units value (1951 of FIG. 19B) for thesame date, then the price is examined. If the price for the same datehas zero discount or a small discount (e.g., less than 1%) and no otherpromotional devices (other than discount) are involved (based onpromotional information provided by the client), then the promotionalflag is set to “1”. For all other dates, if the above-mentionedconditions are not met the promotional flag is set to “0” for thosedates. This set of “0's” or “1's” over the modeled time period defines acrude promotional variable. A simple regression analysis, as is known tothose having ordinary skill in the art, (e.g., a mixed effectsregression) is run on sales volume to obtain a model for predictingsales volume (Step 1905). This analysis will be designed to estimate theimpact on sales volume of: price discount; the crude promotion variable;and other client supplied promotion including, but not limited to,advertisements, displays, and couponing. Using the model a samplecalculation of sales volume is performed (Step 1907). The results of themodel 1953 are compared with the actual sales data 1952 to furtherrefine the promotion flags (Step 1909). If the sales volume isunderpredicted (by the model) by greater than some selected percentage(e.g., 30-50%, preferably 30%) the promotion flag is set to “1” toreflect the effects of a probable non-discount promotional effect. Forexample, if we refer to the region shown as 1954, and the predictedsales volume is 60 units but the actual sales volume was 100 units, themodel has underpredicted the actual sales volume by 40%, greater thanthe preselected level of 30%. Therefore, for that date the promotionflag is set to “1”. This will reflect the likelihood that the increasein sales volume was due to a non-discount promotional effect. Since theremaining modeled results more closely approximate actual salesbehavior, the promotion flags for those results are not reset and remainat “0” (Step 1911). The newly defined promotion flags are incorporatedinto a new model for defining the imputed promotional variable.

i. Imputed Cross-Elasticity Variable

Another useful variable is a cross-elasticity variable. FIG. 20 depictsa flowchart 2000 which illustrates the generation of cross-elasticityvariables in accordance with the principles of the present invention.The generation of an imputed cross-elasticity variable allows theanalysis of the effects of a demand group on other demand groups withinthe same category. Here, a category describes a group of related demandgroups which encompass highly substitutable products and complementaryproducts. Typical examples of categories are, among many others, Italianfoods, breakfast foods, or soft drinks.

An embodiment for generating cross-elasticity variables in accordancewith the principles of the present invention will be illustrated withreference to the following example. The subject category is anabbreviated soft drink category defined by demand groups for diet softdrinks (diet), regular cola soft drinks (reg), caffeine free soft drinks(caff-free), and root beer soft drinks (RB).

The initial dataset information is input into the system (Step 2001).For each demand group the total equivalent sales volume for each storeis calculated for each time period (for purposes of this illustrationthe time period is a week) during the modeled time interval (Step 2003).For each week and each demand group, the average total equivalent salesvolume for each store is calculated for each week over the modeled timeinterval (Step 2005). For each demand group the relative equivalentsales volume for each store is calculated for each week (Step 2007).This may be calculated for each store for each week in accordance withthe formula below:Relative Demand Group Equivalent Sales Volume=Total Equivalent SalesVolume For a Specific Week divided by Average Total Equivalent SalesVolume as Determined For All Weeks in The Modeled Time Interval.

The purpose of the cross-elasticity variable is to quantify the effectsof sales of one demand group upon the sales of another demand group.Therefore, when examining a first demand group, the sales of otherdemand groups within the same category are treated as variables whichaffect the sales of the first demand group. As such, the relative demandgroup equivalent sales volume for the other demand groups is quantifiedand treated as a variable in the calculation of sales volume of thefirst demand group, thereby generating cross-elasticity variables (Step2009). This can be illustrated more easily with reference to the partialdataset illustrated in Tables A and B. These tables reflect one week'sdata (week 1).

TABLE A DEMAND RELATIVE DEMAND GROUP WEEK PRODUCT GROUP EQUIVALENTVOLUME 1 A Diet$\frac{{VolA} + {VolB} + {VolC}}{{avg}.\left( {{VolA} + {VolB} + {VolC}} \right)}$1 B Diet ″ 1 C Diet ″ 1 D Regular$\frac{{VolD} + {VolE} + {VolF}}{{avg}.\left( {{VolD} + {VolE} + {VolF}} \right)}$1 E Regular ″ 1 F Regular ″ 1 G Caff-free$\frac{{VolG} + {VolH} + {VolI}}{{avg}.\left( {{VolG} + {VolH} + {VolI}} \right)}$1 H Caff-free ″ 1 I Caff-free ″ 1 J RB$\frac{{VolJ} + {VolK} + {VolL}}{{avg}.\left( {{VolJ} + {VolK} + {VolL}} \right)}$1 K RB ″ 1 L RB ″

TABLE B DEMAND PRODUCT GROUP CE_(Diet) CE_(Regular) CE_(Caff-free)CE_(RB) A Diet — X X X B Diet — X X X C Diet — X X X D Regular X — X X ERegular X — X X F Regular X — X X G Caff-free X X — X H Caff-free X X —X I Caff-free X X — X J RB X X X — K RB X X X — L RB X X X —

With reference to Table A it is shown that a calculation of RelativeDemand Group Equivalent Volume for product A (a diet soda) is the totalof all equivalent sales volume for the diet soda demand group for thetime period (here week 1). This includes the sum of all equivalent salesvolume for diet soda A, all equivalent sales volume for diet soda B, andall equivalent sales volume for diet soda C. This sum is divided by theaverage sum of equivalent sales volume for diet soda A, diet soda B, anddiet soda C. This Relative Demand Group Equivalent Volume is across-elasticity coefficient (CE_(diet)) for products other than dietsoda (here, regular soda, caffeine-free soda, and root beer). The sametype of calculation is performed with respect to regular soda (reg) and,for that matter, Caffeine-Free (caff-free) and Root Beer (RB) as well.This yields four cross-elasticity coefficients (CE_(diet), CE_(reg),CE_(caff-free), and CE_(RB)). Table B illustrates the relationshipbetween each product, demand group, and the four cross-elasticitycoefficients (CE_(diet), CE_(reg), CE_(caff-free), and CE_(RB)). Thecross-elasticity coefficients are used generate cross-elasticityvariables for each product. In Table B the “-” means the indicatedcross-elasticity coefficient is not applicable to the indicated product.An “x” means the indicated cross-elasticity coefficient is applicable tothe indicated product. For example, if product D (Regular soft drink) isexamined, beginning with Table A, the equation for Relative Demand GroupEquivalent Volume (for product A) is shown. This equation also yieldsthe cross-elasticity coefficient (CE_(reg)) for the demand group regularsoda. Referring now to Table B, the row for product D is consulted.There are “x's” for the coefficients which apply to a determination of across-elasticity variable for product D. Thus, cross-elasticity forproduct D is a function of cross-elasticity coefficients CE_(diet),CE_(caff-free), and CE_(RB). Therefore, the cross-elasticity variablefor product D includes cross-elasticity coefficients CE_(diet),CE_(caff-free), and CE_(RB).

The calculated imputed variables and data are outputted from the imputedvariable generator 304 to the coefficient estimator 308. Some of theimputed variables may also be provided to the financial model engine108.

B. Coefficient Estimator

The coefficient estimator 308 uses the imputed variables and data toestimate coefficients, which may be used in an equation to predictdemand. In a preferred embodiment of the invention, sales for a demandgroup (S) is calculated and a market share (F) for a particular productis calculated, so that demand (D) for a particular product is estimatedby D=S·F. A demand group is defined as a collection of highlysubstitutable products. In the preferred embodiments, the imputedvariables and equations for sales (S) of a demand group and market share(F) are as follows:

1. Modeling Framework

The econometric modeling engine relies on a mixed-model framework,simultaneously utilizing information across all stores and products in aclient category, where a category is defined as a collection ofsubstitutable or complementary products. The mixed model methodology isalso referred to as “Bayesian Shrinkage” Modeling, because by combiningdata from various stores and/or products, one can “shrink” individualparameter estimates towards the average estimate, dampening the extremevalues that would result if traditional regression were used. A basicrationale for this approach is as follows.

In developing product-level volume models for each store within a chain,one may be presented with a wide range of historical data in terms ofmodeling sufficiency. For some stores and/or products, the history willbe quite rich, with many price changes and a variety of promotionpatterns. For other stores and/or products, the history will be quitesparse, with very few price changes and little promotional activity. Tomaximize the stability of estimated model parameters, one might considerdeveloping a single regression model across stores and/or products. Thismodel might have stable parameter estimates; however, it would not fullyleverage the store and/or product variability, and the resulting modelwould likely not predict well for a particular store and/or product. Onthe other hand, one might consider developing individual regressionmodels for each store and/or product, to utilize the specificinformation contained in the data history. While these models might fitand predict well for the stores and/or products with substantial pricevariability, models would not be estimable for stores and/or productswithout a rich data history.

A mixed-model framework addresses the need for both highly predictivemodels and the existence of an estimable model for each store andproduct. In a mixed-effect model, information (in the form of datahistory) is leveraged across all stores and products, and a singlecohesive model is built. Stores and products with little or noinformation in their data history default to an “average” (fixed-effect)model. Likewise, stores and products with a wealth of information intheir data history will end up with unique parameter estimates tailoredto their response pattern, via estimation of non-zero store and/orproduct-specific adjustment factors (random effects) which are added tothe fixed-effect portion of the model.

2. Terminology

The equivalent price of a product is defined as the price of astandardized unit of measure, which may be calculated based on theproduct description and the spread of sizes/counts that apply to thatdescription. Each individual product price is divided by thisstandardized unit of measure to obtain the equivalent price.

A demand group is defined as a set of products that are substitutes ornear substitutes for each other. A product can belong to only one demandgroup. A product category consists of one or more demand groups. Forthis example, attention is restricted to a single category consisting ofmultiple demand groups.

Both models:

Subscript i: Demand group (primary). A demand group is a collection ofhighly substitutable products.

Subscript j: Demand group (secondary). A secondary demand group isanother demand group in the same category as the primary demand group,where a category is defined as a collection of substitutable orcomplementary products.

Subscript k: Product, where products are items with common UPC numbers.

Subscript t: Time period, which may be days, weeks, or hours.

Subscript B: Baseline, which is a state of product if there was nopromotion.

Subscript n: Number of time periods away from current time period.

ε: Error term for regression equations, with appropriate subscripts percontext.

3. Stage 1 (Sales) Model

a. Sales Model Multipliers (Data Values, or Covariates) and DependentVariables

S_(i,t): The equivalent sales of demand group i in period t in store sin dollars. Equivalent sales may be defined as sales of equivalent unitsof products being compared.

S_(Bl,t): The equivalent baseline sales of demand group i in store s inperiod t.

S _(l,t): The equivalent sales of demand group i in store s averagedover periods leading up to period t.

R_(i,t): The equivalent revenue of demand group i in period t.

R_(Bl,t): The equivalent baseline revenue of demand group i in period t,which would be baseline sales times the baseline price.

R _(i,t): The equivalent revenue of demand group i averaged over periodsleading up to period t.

P_(l,t): The equivalent price of demand group i at store s in timeperiod t, calculated as total equivalent revenue of demand group idivided by total equivalent sales in the period (S_(i,t)/R_(l,t)), andwhere the equivalent price is the price of an equivalent unit, such as aprice per ounce.P _(l,t): The average equivalent price of demand group i in store s fortime period t, calculated as average total equivalent revenue of demandgroup i divided by average total equivalent sales ( S _(l,t)/ R _(l,t)).P _(l,t): The average competitor (e.g. a competing store in the area)equivalent price of demand group i in store s for time period t,calculated as average competitor total equivalent revenue of demandgroup i divided by average competitor total equivalent sales.M_(i,t): The promotion level for demand group i in store s in period t.X_(l,t): The seasonality index for demand group i in store s in periodt.TS_(t): The total dollar sales for the entire store in period t,computed using historical data.T S _(t): The total dollar sales for the region in period t, computedusing historical data. A region would be a grouping of stores possiblyin a certain geographical area.

b. Sales Model Factors (parameters to be estimated)

γ₁: The price elasticity factor for demand group i measured with respectto the deviations of the weighted average price of the demand group fromthe past weighted average price of the group. It measures thesensitivity of sales of equivalized units of the demand group withrespect to the group price.ν₁: The promotion factor for demand group i. This factor measures thesensitivity of the equivalent sales of the demand group to the generalpromotion level of the group. ψ_(l): The seasonality factor for demandgroup i. This factor measures the sensitivity of the equivalent sales ofthe demand group to seasonality.κ₁: The seasonality-price interaction factor that measures theinteraction of weighted average price deviations and seasonality fordemand group i. The seasonality and the group price may interact witheach other in a nonlinear way. This factor measures a degree ofnonlinearity.δ_(l,n): The time lag factor for demand group i and delay of n weeks.The time lag factor measures the level of forward buying or stockpilingactivity by customers. Note that this is the only factor that isestimated at the demand group level rather than the store-demand grouplevel.Φ_(l,j): The cross elasticity factor for demand group i and demand groupj. This factor measures how sales of a demand group are affected by thesales of other demand groups in the same category.η_(l,n): The competitive price factor for demand group i measured withrespect to the difference between the weighted average price of thedemand group within the store and outside competitors. This factormeasures the effect of competitive activity on the sales of products ina given demand group.π_(l): The traffic factor for demand group i. Sales may be affected bythe overall traffic through a store. This factor quantifies therelationship.θ_(i): The day-of-week (DOW) effect for demand group i. Each day of aweek could have a different sales pattern. This factor quantifies therelationship.K_(l): The constant (intercept) associated with demand group i.

C. The sales model is:

$\begin{matrix}{{\ln\left( \frac{{\hat{S}}_{i,t}}{S_{{Bi},t}} \right)} = {{\hat{K}}_{i} + {{\hat{\gamma}}_{1}\frac{P_{i,t}}{{\overset{\_}{P}}_{i,t}}} + {{\hat{v}}_{i}M_{i,t}} + {{\hat{\psi}}_{i}X_{i,t}} + {{\hat{\kappa}}_{i}X_{1,t}\frac{P_{i,t}}{{\overset{\_}{P}}_{i,t}}} + {\sum\limits_{n = 1}^{\tau}{{\hat{\delta}}_{i,n}\frac{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}S_{i,r}}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}{\overset{\_}{S}}_{i,r}}}} + {\sum\limits_{j \neq i}{{\hat{\phi}}_{i,j}\frac{{\hat{S}}_{j,t}}{{\overset{\_}{S}}_{j,t}}}} + {{\hat{\eta}}_{i,t}\left( \frac{{\overset{\_}{P}}_{i,t} - {\overset{\_}{\overset{\_}{P}}}_{i,t}}{{\overset{\_}{\overset{\_}{P}}}_{i,t}} \right)} + {{\hat{\pi}}_{i}\frac{{TS}_{t}}{T{\overset{\_}{S}}_{t}}} + {{\hat{\theta}}_{i}\frac{S_{i,{t - 7}}}{{\overset{\_}{S}}_{i,{t - 7}}}}}} & {{Equation}\mspace{20mu} 1}\end{matrix}$

In the above model (Equation 1), the dependent variable, demand groupequivalent sales, is indexed (divided) by baseline demand groupequivalent sales to provide a type of normalization. This normalizingallows easier comparison within a demand group and between demandgroups. If a reasonable approximation of baseline demand group salescannot be imputed, the dependent variable may alternatively be indexedby demand group equivalent sales averaged over a specified number oftime periods prior to the current one ( S _(l,t)).

In the time lag term, τ represents the number of “time buckets”preceding a time period that will be included in the model, and mrepresents the size of the “time bucket,” in number of time periods.

Inclusion of several covariates (day-of-week, store traffic,within-market competitive pricing) is contingent upon the time dimensionand scope of available client data. Therefore, if data is reported on aweekly basis, so that there is no data according to day of the week, theday of the week parameters will not be included.

d. Sales Model Unbiasing Factor

In regression calculations, returning to the original scale after alogarithmic transformation creates a bias. To correct for this bias, theBaskersville's method is used which consists of adding an unbiasingfactor to the equation. This factor is the mean square error({circumflex over (σ)}²) of the sales model divided by 2.

The equation for predicting demand group sales is thus:

$\begin{matrix}{\left( \frac{{\hat{S}}_{i,t}}{S_{{Bi},t}} \right) = {\exp\left( \begin{matrix}{{\hat{K}}_{i} + {{\hat{\gamma}}_{i}\frac{P_{i,t}}{{\overset{\_}{P}}_{i.t}}} + {{\hat{v}}_{i}M_{i,t}} + {{\hat{\psi}}_{i}X_{i,t}} + {{\hat{\kappa}}_{i}X_{i,t}\frac{P_{i,t}}{{\overset{\_}{P}}_{i,t}}} + {\sum\limits_{n = 1}^{\tau}{{\hat{\delta}}_{i,n}\frac{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}S_{i,r}}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}{\overset{\_}{S}}_{i,r}}}} + {\sum\limits_{j \neq i}{{\hat{\phi}}_{i,j}\frac{{\hat{S}}_{j,t}}{{\overset{\_}{S}}_{j,t}}}} +} \\{{{\hat{\eta}}_{i,t}\left( \frac{{\overset{\_}{P}}_{i,t} - {\overset{\_}{\overset{\_}{P}}}_{i,t}}{{\overset{\_}{\overset{\_}{P}}}_{i,t}} \right)} + {{\hat{\pi}}_{i}\frac{{TS}_{t}}{T{\overset{\_}{S}}_{t}}} + {{\hat{\theta}}_{i}\frac{S_{i,{t - 7}}}{{\overset{\_}{S}}_{i,{t - 7}}}} + \frac{{\hat{\sigma}}^{2}}{2}}\end{matrix} \right)}} & {{Equation}\mspace{20mu} 2}\end{matrix}$

4. Stage 2 (Share) Model

a. Share Model Multipliers (Data Values, or Covariates) and DependentVariables

F_(l,k,t): The fraction of demand group i equivalent sales comprised byproduct k in time period t (market share of product k).

F _(l,·,t): The average fraction of demand group i equivalent sales withrespect to time period t. To allow linear modeling of the regressionequation, this data value is used to provide centering.

P_(Bl,k,t): The equivalent base price of product k in demand group i intime period t.

P _(Bl,(k),t): The average equivalent base price of all products otherthan product k in demand group i for time period t.

P_(R Bl,k,t): The relative equivalent base price of product k in demandgroup i for time period

${t\left( {= \frac{P_{{Bi},k,t}}{{\overset{\_}{P}}_{{Bi},{(k)},t}}} \right)}.$P _(R Bi,·,t): The average relative equivalent base price in demandgroup i for time period t.M_(p,l,k,t): The level of promotion type p (kind of promotion) forproduct k in demand group i in time period t. There can be up to n_(p)promotion factors estimated in the model.M _(p,l,·,t): The average level of promotion type p in demand group ifor time period t.

b. Share Model Factors (Parameters to be Estimated)

ρ_(l,k): The relative base price elasticity factor for product k indemand group i.

σ_(p,l,k): The promotion factor p for product k in demand group i. Therecan be up to n_(p) promotion factors estimated in the model.

χ_(l,k,n): The time lag factor for product k in demand group i and delayof n weeks.

Λ_(l,k): The constant (intercept) associated with product k in demandgroup I.

The model for predicting product share (market share) is:

$\begin{matrix}{{\hat{F}}_{i,k,t} = \frac{\begin{matrix}{\exp\left\{ {{\hat{\Lambda}}_{i,k} + {{\hat{\rho}}_{i,k}\left( P_{{Ri},k,t} \right)} + {\sum\limits_{p = 1}^{n_{p}}{{\hat{\sigma}}_{p,i,k}\left( M_{p,i,k,t,} \right)}} +} \right.} \\\left. {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\left( F_{i,k,r} \right)}}} \right\}\end{matrix}}{\begin{matrix}{\sum\limits_{l \in {Dem}_{i}}{\exp\left\{ {{\hat{\Lambda}}_{i,l} + {{\hat{\rho}}_{i,l}\left( P_{{Ri},l,t} \right)} + {\sum\limits_{p = 1}^{n_{p}}{{\hat{\sigma}}_{p,i,l}\left( M_{p,i,l,t,} \right)}} +} \right.}} \\\left. {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\left( F_{i,l,r} \right)}}} \right\}\end{matrix}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

This model calculates demand for a product divided by demand for thedemand group of the product.

The product intercept Λ_(l,k) is not individually estimated in themodel. Instead, each product is characterized according to productattributes, such as brand, size group (small/medium/large), form,flavor, etc. A store-specific estimate of the effect corresponding toeach attribute level effects is obtained, and each product intercept isthen constructed by summing over the applicable attribute levelestimates.

Thus,

${{\hat{\Lambda}}_{i,k} = {\sum\limits_{a = 1}^{n_{a}}{\sum\limits_{b = 1}^{n_{b{(a)}}}{{\hat{\xi}}_{a_{b}} \cdot I_{k,a_{b}}}}}},$whereξ_(a) _(b) is the effect of attribute a, level b, andI_(k,a) _(b) is an indicator variable for product k,

$= \begin{Bmatrix}{1,} & {{if}\mspace{14mu}{product}\mspace{14mu}{has}\mspace{14mu}{level}\mspace{14mu} b\mspace{14mu}{of}\mspace{14mu} a} \\{0,} & {else}\end{Bmatrix}$

The attribute values may be used to predict sales of new products withvarious attribute combinations.

5. Linearization of Multinomial Logistic Equation

The multinomial logistic function that defines the market shareequations for each product is nonlinear but there exist standardtechniques to transform it to a linear function instead, which may makemodeling easier. An example of a transformation to a linear function isas follows:

In this section the store index is ignored.

For a particular time period t:

${{{Let}\mspace{14mu} F_{i,k}} = \frac{\exp\left( {\alpha_{i,k} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},k}}} + ɛ_{i,k}} \right)}{\sum\limits_{j = 1}^{k}{\exp\left( {\alpha_{i,1} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},j}}} + ɛ_{i,j}} \right)}}},$where α_(l) is the intercept for demand group i, β are the covariates,and P is the number of covariates in the share model

$\;{{\log\left( F_{i,k} \right)} = {\alpha_{i,k} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},k}}} + ɛ_{i,k} - {\log\left( {\sum\limits_{j = 1}^{k}{\exp\left( {\alpha_{i,j} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},j}}} + ɛ_{i,j}} \right)}} \right)}}}$${{Let}\mspace{14mu}{\log\left( {\overset{\sim}{F}}_{i} \right)}} = {{\frac{1}{k}{\sum\limits_{j = 1}^{k}{\log\left( F_{i,k} \right)}}} = {{\overset{\_}{\alpha}}_{i} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot {\overset{\_}{X}}_{pi}}} + {\overset{\_}{ɛ}}_{i} - {\log\left( {\sum\limits_{j = 1}^{k}{\exp\left( {\alpha_{i,j} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},j}}} + ɛ_{i,j}} \right)}} \right)}}}$${Thus},{{\log\left( \frac{F_{i,k}}{{\overset{\sim}{F}}_{i}} \right)} = {{\alpha_{i,k} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},k}}} + ɛ_{i} - {\log\left( {\sum\limits_{j = 1}^{k}{\exp\left( {\alpha_{i,j} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},j}}} + ɛ_{i,j}} \right)}} \right)} - {\overset{\_}{\alpha}}_{i} - {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot {\overset{\_}{X}}_{pi}}} - {\overset{\_}{ɛ}}_{i} + {\log\left( {\sum\limits_{j = 1}^{k}{\exp\left( {\alpha_{i,j} + {\sum\limits_{p = 1}^{P}{\beta_{p} \cdot X_{{pi},j}}} + ɛ_{i,j}} \right)}} \right)}} = {\left( {\alpha_{i,k} - {\overset{\_}{\alpha}}_{i}} \right) + {\sum\limits_{p = 1}^{P}{\beta_{p}\left( {X_{{pi},k} - {\overset{\_}{X}}_{pi}} \right)}} + \left( {ɛ_{i,j} - {\overset{\_}{ɛ}}_{i}} \right)}}}$

To model share in a linear framework, we simply center all covariatesand model log(F_(1,k))−log({tilde under (F)}_(i)), where {tilde under(F)}_(i) is geometric mean of F_(i,k):

${{\log\;\left( F_{i,k,t} \right)} - {\log\left( {\overset{\sim}{F}}_{i,\bullet,t} \right)}} = {\Lambda_{i,k,t} + {\rho_{1,k}\left( {P_{{R1},k,t} - {\overset{\_}{P}}_{{R1},\bullet,t}} \right)} + {\sum\limits_{p = 1}^{n_{p}}\;{\sigma_{p,i,k}\left( {M_{p,i,k,t} - {\overset{\_}{M}}_{p,i,\bullet,t}} \right)}} + {\sum\limits_{n = 1}^{\tau}\;{\chi_{i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\;\left( {F_{i,k,r} - {\overset{\_}{F}}_{i,\bullet,r}} \right)}}}}$

C. Combined (Product Sales) Model

The model for predicting sales for product k in demand group i in timeperiod t is thus given by:

{circumflex over (D)}_(i,k,t)={circumflex over (F)}_(l,k,t)Ŝ_(l,k)

III. FINANCIAL MODEL ENGINE

The financial model engine 108 receives data 132 from the stores 124 andmay receive imputed variables (such as baseline sales and baselineprices) and data from the econometric engine 104 to calculate fixed andvariable costs for the sale of each item. To facilitate understanding,FIG. 5 is a block diagram to illustrate some of the transaction coststhat occur in retail businesses of a chain of stores. The chain ofstores may have a headquarters 504, distribution centers 508, and stores512. The headquarters 504 may place an order 516 to a manufacturer 520for goods supplied by the manufacturer 520, which generates an orderplacement cost. The manufacturer 520 may ship the goods to one of thedistribution centers 508. The receiving of the goods by the distributioncenter 508 generates a receiving cost 524, a cost for stocking the goods528, and a cost for shipping the goods 532 to one of the stores 512. Thestore 512 receives the goods from one of the distribution centers 508 orfrom the manufacturer 520, which generates a receiving cost 536 and acost for stocking the goods 540. When a customer purchases the item, thestores 512 incur a check-out cost 544. With the large number of retailchains, different purchasing and delivery processes may be used. Evenwithin a single chain, different manufacturers may provide differentinvoicing and delivery procedures and costing system, which may bedifferent than the processes illustrated in FIG. 5.

The financial model engine 108 should be flexible enough to provide acost model for these different procedures. These different costs mayhave variable cost components where the cost of an item is a function ofthe amount of sales of the item and fixed cost components where the costof an item is not a function of the amount of sales of the item. Thefinancial model engine 108 uses these fixed and variable costs todetermine C_(s,i,k,t), where C_(s,l,k,t), is a cost for a particularproduct (k) given a store (s), demand group (i), and a day (t). Thefinancial model engine 108 uses industry data to provide standardestimates. For example, instead of measuring how long it takes to stocka box of an item, an industry data may be used to estimate this time.The standard estimates helps to reduce the amount of data that must becollected. In a preferred embodiment of the invention, the stores mayonly need to supply labor costs of the stores and distribution centers,cost of capital, size of an item, and number of items in a case to allowa cost modeling. Likewise, the preferred embodiment may infer the amountof shelf-space an item utilizes from the cubic feet of the item, thevolume of sales of the item, and how often the store is replenished withthe item. By using these estimations, costs may be more easilycalculated on a store level, instead of being averaged over all of thestores. This is because if large amounts of data are measured by hand,measuring such data for each store would be difficult. The tailoring ofcosts per store allows the maximization of profits for each store. In anexample of the preferred embodiment of the invention, the financialmodel engine 108 comprises an activity-based costing module that usesthe parameters and calculations as follows:

A. Definitions

The Activity-based costing module computes variable and fixed costs forproducts at specific store locations. The ABC modeling componentintroduces the Distribution Center entity to track costing activities atretailers' distribution centers and regional stores.

Some numbers related to properties of the product. For each product aCasePack number is provided, which is the number of products in a case.A CaseCube number provides the cubic feet of each case of the product.

Certain specifiers are used to determine costs related to a distributioncenter. A ReceivingMethod specifier allows the specification ofreceiving the items by pallet or by individual cases, which specifydifferent stocking costs. A ShelfStockMethod specifier allows thespecification of the type of method that may by used to stock a shelf.For example, with baby food with many individual bottles, if the caseprovides a tray, which allows all bottles in the case to be quickly slidonto the shelf, then the stocking costs would be less than for a casewithout a tray, where all bottles must be manually lifted out of thecase and placed on the shelf. If an entire box or case of an item may beplaced on a shelf, the shelf stocking costs would also be less. ACaseListCost specifier specifies the wholesale price for a case. TheCaseAllowance specifier specifies any discount off of the wholesaleprice of a case. The AllowBackhaul specifier allows the specification ofa deduction provided by a manufacturer when a store is near themanufacturer so that on a return trip to a distribution center the storehauls the goods from the manufacturer. The BackhaulPerCase is the amountof the back haul discount per case. The VendorDaysCredit specifierallows the specification of the number of days of credit a manufacturermay give after shipment. The InventoryInDays specifier allows thespecification of the number of days that the product is held asinventory at the distribution center.

If a wholesaler or manufacturer ships the product directly to the store,then different cost factors may be required, while distribution centerrelated costs may be eliminated. The DropShipMethod specifier allows thespecification of whether the directly shipped product is directlyshipped to the shelf, a store back room or to a retailers distributioncenter.

The AvgWklyCase specifier specifies the average weekly number of casesof product sold by a store. This would be the average number of units ofproduct sold by the store divided by the number of units of product percase.

Various costs may be associated with the process of having a productshipped directly to a store. The DropShipMethod specifier is used tospecify whether a manufacturer delivers a product to a store shelf, orto a store back room, or to a distribution center.

Various labor costs may be associated with different distributioncenters and stores. The DCLabor Rate is a specifier used to specify theaverage labor rate at each distribution center. The StoreStockLabor Rateis a specifier used to specify the average labor rate of a stock clerkat a store. The StoreCheckoutLabor Rate is a specifier used to specifythe average labor rate of a cashier at a store. Each of these labor rateaverages may be different between different stores and differentdistribution centers. Each average per store or per distribution centermay be used in the database.

A ProductStorageType specifier allows the specification of dry warehousestorage for dry items, or refrigerated storage, or frozen storage, whicheach have different costs. A DCDryCostPFT specifier is used to specifythe cost of storing dry items, per cubic foot at the distributioncenter. This cost may be provided by the distribution center, but anindustry average may be used as a default value. A DCRefrigCostPFTspecifier is used to specify the cost of storing refrigerated items, percubic foot at the distribution center. This cost may be provided by thedistribution center, but an industry average may be used as a defaultvalue. A DCFrozenCostPFT specifier is used to specify the cost ofstoring frozen items, per cubic foot at the distribution center. Thiscost may be provided by the distribution center, but an industry averagemay be used as a default value. A StoreDryCostPFT specifier is used tospecify the cost of storing dry items, per cubic foot at the store. Thiscost may be provided by the store, but an industry average may be usedas a default value. A StoreRefrigCostPFT specifier is used to specifythe cost of storing refrigerated items, per cubic foot at the store.This cost may be provided by the store, but an industry average may beused as a default value. A StoreFrozenCostPFT specifier is used tospecify the cost of storing frozen items, per cubic foot at the store.This cost may be provided by the store, but an industry average may beused as a default value.

A DCCubeSecs specifier specifies the number of seconds it takes tohandle a cubic foot of product at the distribution center. This time maybe provided by the distribution center or an industry average may beused as a default value. A DCCaseUnitizedSecs specifier specifies theaverage number of seconds required to handle a palletized load at thedistribution center. This time may be provided by the distributioncenter or an industry average may be used as a default value. ADCCaseDeadPileSecs specifier specifies the average number of secondsrequired to handle a non-palletized load (a dead pile) at thedistribution center. This time may be provided by the distributioncenter or an industry average may be used as a default value.

A StoreCubeNTSecs specifier specifies a time in seconds to handle anon-tray cubic foot of product at a store. This time may be provided bythe store or an industry average may be used as a default value. AStoreCaseNTSecs specifier is used to specify a time in unitized secondsto handle a non-tray case of a product at a store. This time may beprovided by the store or an industry average may be used as a defaultvalue. A StorePkgNTSecs specifier is used to specify a time in unitizedseconds to handle a non-tray package of a product at a store. This timemay be provided by the store or an industry average may be used as adefault value. So in a case with twenty four packages, when a package ispurchased one twenty fourth of a case is purchased, which may have avolume of a tenth of a cubic foot.

A StoreCubeTraySecs specifier specifies a time in seconds to handle acubic foot of a tray product at a store. This time may be provided bythe store or an industry average may be used as a default value. AStoreCaseTraySecs specifier is used to specify a time in unitizedseconds to handle a case of a tray product at a store. This time may beprovided by the store or an industry average may be used as a defaultvalue. A StorePkgTraySecs specifier is used to specify a time inunitized seconds to handle a package of a tray product at a store. Thistime may be provided by the store or an industry average may be used asa default value.

A StoreCheckoutPkgSecs specifier specifies the time it takes to checkout a package of product. This time may be provided by the store or anindustry average may be used as a default value. An AvgDeliveryDistanceis an average distance for shipping from a distribution center to astore. A TruckCubic specifier specifies the number of cubic feet thatmay be loaded on a truck. A StoreLabor Pct specifier specifies thepercentage of store labor that is required for the different deliverymethods. If a distribution center delivers an item to a store, then theStoreLabor Pct is 100%, since such products must be fully processed. Ifa vendor delivers a product directly to a backroom of the store, then aStoreLabor Pct of 75% may be used, since the store does not need to doas much work. If a vendor delivers a product directly to a store shelf,then a StoreLabor Pct of 25% may be used, since the store may be allowedto do less work for such deliveries.

An AnnualOpptyCost specifier specifies the annual opportunity cost ofthe inventory in the distribution centers and stores. Annual opportunitycost may be calculated by determining the inventory in the distributioncenters and the inventory in the stores to determine the dollar value ofthe inventory and then subtract the dollar value of inventory that is oncredit and multiplying the difference by the cost of capital percentage.Generally, the inventory may be calculated from the raw data, so thatthe customer only needs to provide the cost of capital to determineannual opportunity cost. An InvoiceProcessing specifier specifies thecost of providing an invoice (placing an order for a product with avendor) per case. A Transportation specifier specifies a cost oftransporting an item per mile. A ShoppingBag specifier specifies thecost of a shopping bag.

A database would provide an identification code for each distributioncenter and would provide for the specification of the location of eachdistribution center. Similarly, each store would have an identificationcode and the database would also allow the specification of the locationof each store. A DeliveryFrequency specifier allows the specification ofthe number of days between deliveries to a store.

B. Cost Calculation

To calculate C_(s,i,k,t) (the cost of a product (i) in a demand group(k) in a store (s) at a time (t) fixed and variable cost components arecomputed from the given data structures, as follows:

The bag cost calculation computes total shopping bag costs for a productin a distribution center's region.

1. Bag Costs

The bag cost calculation is:BagCost=(DistributionCenter.ShoppingBag/0.6)*Product.CaseCube/Product.CasePack,where one bag in this example is 0.6 cubic feet, so that bag cost is thecost of a bag divided by the volume of a bag, the quantity times thevolume of a case of a product divided by the number of products in acase. The distribution center identifier used in the equation is so thatif bags have different costs at different distribution centers, thedifferent cost may be noted by the distribution center identifier.

2. Checkout Labor Costs

The checkout labor costs is computed as cost per package from the rateat which items are checked out and the checkout clerk's pay rate. TheCheckout Labor Cost Calculation is:CheckoutCostPerPkg=StoreCheckoutPkgSecs*StoreCheckoutLaborRate/3600,so that the Checkout Labor cost is the product of the store check outrate in packages per second and the store labor rate in dollars per hourdivided by 3600 seconds per hour.

3. Distribution Center Inventory Costs

The Distribution Center Inventory Costs computes the inventory cost perpackage at a given distribution center. If a product is not stored at adistribution center, then the inventory cost is zero. If a product canbe backhauled, the inventory cost is reduced by the backhaul rate. TheDistribution Center Inventory Cost Calculation is as follows:DCInvCostPerPkg=ISNULL(ProductDistributionCenter.InventoryInDays, 0)*(DistributionCenter.AnnualOpptyCost/365)*(Product.CaseListCost−Product.CaseAllowance−IIf(Product.AllowBackHaul=1,Product.BackhaulPerCase, 0))/Product.CasePack.

The zero in the first factor allows the cost to be zero if directshipping is used. Otherwise, the inventory costs are the inventory costsfor a distribution center times the annual opportunity cost for thedistribution center divided by 365 days times the cost per case minusthe allowances such as a product case allowance and back haul, with theback haul discount being zero if there is no back haul, where theproduct is divided by the number of products per case or package.

4. Distribution Center Labor Costs

The Distribution Center's Labor Costs per package is computed dependingupon the distribution center's receiving method. If a product is notshipped from the distribution center, the cost is zero. The calculationof labor costs at the distribution center calculates the time it takesto process a product at the distribution center and multiplies it by alabor rate. An example of an algorithm for calculating distributioncenter labor costs is as follows:

Distribution Center Labor Cost CalculationIf Product. ReceivingMethod=1 ThenDCLaborCostPerPkg=((DistributionCenter.DCCubeSecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.DCCaseUnitizedSecs/Product.CasePack))*(DistributionCenter.DCLaborRate/3600)ElseDCLaborCostPerPkg=((DistributionCenter.DCCubeSecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.DCCaseDeadPileSecs/Product.CasePack))*(DistributionCenter.DCLabor Rate/3600)

In the first branch of the algorithm for a unitized (palletized) load,first the number of seconds to process a cubic foot of product processedby the distribution center is multiplied by the number of cubic feet percase divided by the number of products per case to obtain the number ofseconds to process a product at the distribution center. The number ofseconds to process a unitized (palletized) case by the distributioncenter divided by the number of products per case yields the number ofseconds for each product being processed by the distribution center.These two products are added to obtain a total number of second toprocess a product processed by the distribution center. This total ismultiplied by the distribution center labor rate per hour divided by3600 seconds per hour to obtain the distribution center labor cost forpalletized product.

In the second branch of the algorithm for a dead pile (non-palletized)load, first the number of seconds to process a cubic foot of productprocessed by the distribution center is multiplied by the number ofcubic feet per case divided by the number of products per case to obtainthe number of seconds to process a product at the distribution center.The number of seconds to process a dead pile (non-palletized) case bythe distribution center divided by the number of products per caseyields the number of seconds for each product being processed by thedistribution center. These two products are added to obtain a totalnumber of seconds to process a product processed by the distributioncenter. This total is multiplied by the distribution center labor rateper hour divided by 3600 seconds per hour to obtain the distributioncenter labor cost for non-palletized products.

5. Distribution Center Calculations

The Distribution Center figures are aggregated by Product Type.

a. Sum of Average Weekly Cases

The SumOfAvgWklyCases specifier specifies the total of all averageweekly cases of all stores for a particular product for a particulardistribution center. This specifier may be calculated as follows:SumOfAvgWklyCases=SUM(ProductLocDC.AvgWklyCases)

b. Count of Product Location Distribution Centers

The CountProdLocDC specifier specifies the number of stores perdistribution center that sell the product.

c. Average Distribution Center Inventoried Days

The AvgOfDCInvDays specifier specifies the average number of days ofinventory of a particular product at a distribution center. Thedistribution center may provide this number or an industry standard maybe used as a default value.

d. Average Product Case Cubes

The AvgOfCaseCube specifier specifies an average cubic foot volume of acase of a product. This average may use an industrial standard as adefault if a volume of a case of a product is not provided.

e. Distribution Center Space

The DCSpace specifier is a calculation of the volume of space of aproduct in the inventory of a distribution center. The calculation forDCSpace is as follows:DCSpace=(AvgOfDCInvDays/7)*SumOfAvgWklyCase* AvgOfCaseCube

The DCSpace may be the product of the Average Distribution CenterInventoried Days divided by seven (to convert days into weeks) times theSum of the Average Weekly Cases of the product times the Average ProductCase Cube to yield the volume.

f. Distribution Center Cube

The DC_Cube specifier specifies the volume of the distribution centerallocated to the product. The distribution center may be divided intoslots. In this embodiment, only whole numbers of slot spaces may beallocated to a product. In this example, the slot space is 80 cubicfeet. Therefore, the volume used by inventory must be rounded up to thenext 80 cubic feet increment. In such an example, an algorithm forcalculating the number of slots allocated to the product is as follows:

IfDC_Space >0 Then

DC_Cube=Int((DC_Space/80)+0.999)*80

Else

DC_Cube=0

In the first branch, if the Distribution Center Space is greater thanzero, then the Distribution Center Space is divided by 80 cubic feet andthe result is added to 0.999. The integer value of the resulting sum isused, so that the non-integer part of the sum is truncated. The integervalue is multiplied by 80 cubic feet to convert the number of slots thatare allocated to the product to a volume.

In the second branch, if the Distribution Center Space is equal to zero,no slots are allocated to the product.

g. Distribution Center Space Cost per Store

The DCSpaceCostPerStore specifier specifies distribution center spacecost per store. In this embodiment, an algorithm to calculateDCSpaceCostPerStore, where dry goods are ProductStorageType 1,refrigerated goods are ProductStorageType 2 and frozen goods areProductStorageType 3, is as follows:If ProductStorageType=1 ThenDCSpaceCostPerStore=DC_Cube*DCDryCostPFT/(Product.CasePack*52*CountProdLocDCIf ProductStorageType=2 ThenDCSpaceCostPerStore=DC_Cube*DCRefrigCostPFT/(Product.CasePack*52*CountProdLocDC)If ProductStorageType=3 ThenDCSpaceCostPerStore=DC_Cube*DCFrozenCostPFT/(Product.CasePack*52*CountProdLocDC)

In the first branch of the algorithm for dry goods, first theDistribution Center Cube is multiplied by the DCDryCostPFT. This productis then divided by the product of the number of product per case times52 weeks in a year times the number of stores per distribution center toobtain the Distribution Center Space Cost Per Store.

In the second branch of the algorithm for refrigerated goods, first theDistribution Center Cube is multiplied by the DCRefrigCostPFT. Thisproduct is then divided by the product of the number of product per casetimes 52 weeks in a year times the number of stores per distributioncenter to obtain the Distribution Center Space Cost Per Store.

In the third branch of the algorithm for refrigerated goods, first theDistribution Center Cube is multiplied by the DCFrozenCostPFT. Thisproduct is then divided by the product of the number of product per casetimes 52 weeks in a year times the number of stores per distributioncenter to obtain the Distribution Center Space Cost Per Store.

6. Invoice Processing

The Invoice Processing calculation computes the Invoice processing costper Product. To calculate Invoice Processing (HQInvPro), the cost ofInvoice Processing per case is divided by the number of products percase, as follows:HQInvProc=InvoiceProcessing/Product.CasePack

7. Product Location Space

The Product Location (store) Space calculation computes a store'savailable space for a product.

a. Average Delivery Frequency Calculation

The AvgDeliveryFreq specifier specifies the average number of deliveriesto a store per week averaged over several weeks. An equation tocalculate AvgDeliveryFreq is:AvgDeliveryFreq=AVG(Location.DeliveryFrequency)

b. Sum of Average Weekly Cases Calculation

The SumOfAvgWklyCases specifier is the sum of the AvgWklyCases overseveral weeks.SumOfAvgWklyCases=SUM(.AvgWklyCases)

c. Average Case Cube Calculation

The AvgCaseCube specifier specifies the average volume of a case of aproduct.

d. Location (Store) Cube Calculation

The LocationCube specifier specifies the volume of the store (location)that the product takes up on the shelf and/or back room. An equation forcalculating LocationCube may be as follows:LocationCube=(0.5*Product.CaseCube)+(Product.CaseCube*AvgWklyCases/DeliveryFrequency)

A volume of half a case is the first term. The second term is the volumeof a case times the average number of cases delivered weekly divided bythe frequency of deliveries per week, which estimates the volume ofinventory delivered to the store with each delivery. So this sumestimates that the store will need a volume of about the volume for eachdelivery plus half a case to be used for the product.

e. Location Space Cost Calculation

After determining the volume of a product in the store, the store spacecost (LocationSpaceCost) may be calculated. This cost is dependent uponwhether the product is a dry item ProductStorageType=1, a refrigerateditem ProductStorageType=2, or a frozen item ProductStorageType=3.If ProductStorageType=1 ThenLocationSpaceCost=LocationCube*StoreDryCostPFT/(Product.CasePack *52)If ProductStorage Type=2 ThenLocationSpaceCost=LocationCube*StoreRefrigCostPFT/(Product.CasePack*52)If ProductStorageType=3 ThenLocationSpaceCost=LocationCube*StoreFrozenCostPFT/(Product.CasePack*52)

In the first branch of the algorithm for dry goods, first the Location(store) Cube is multiplied by the StoreDryCostPFT to obtain a storestorage cost per week. This product is then divided by the product ofthe number of product per case times 52 weeks in a year to obtain theLocation Space Cost, which is the store storage cost per week per itemof product.

In the second branch of the algorithm for refrigerated goods, first theLocation (store) Cube is multiplied by the StoreRefrigCostPFT to obtaina store storage cost per week. This product is then divided by theproduct of the number of product per case times 52 weeks in a year toobtain the Location Space Cost, which is the store storage cost per weekper item of product.

In the third branch of the algorithm for refrigerated goods, first theLocation (store) Cube is multiplied by the StoreFrozenCostPFT to obtaina store storage cost per week. This product is then divided by theproduct of the number of product per case times 52 weeks in a year toobtain the Location Space Cost, which is the store storage cost per weekper item of product.

8. Transportation Cost Per Package

The Transportation Cost Per Package (TransCostPerPkg) computes the costto truck products to stores. Transportation Cost Per Package may becalculated as follows:TransCostPerPkg=Product.CaseCube/(TruckCubic*Product.CasePack)*Transportation*AvgDeliveryDistance

The volume of the case is divided by the product of the volume of thetruck times the number of items in each case. The result is multipliedby the cost per mile of delivery times the average delivery distance.

9. Product Location (Store) Inventory Cost

The Product Location Inventory Cost calculation computes a store'sinventory cost for a product. An equation for calculating ProductLocation Inventory Cost may be as follows:ProductLocInvCost=(((LocationCube*0.75/Product.CaseCube)/SumOfAvgWklyCases)−(VendorDaysCredit/7))*((Product.CaseListCost−Product.CaseAllowance−IIf(Product.AllowBackhaul=1,Product.BackhaulPerCase, 0))/Product.CasePack)*AnnualOpptyCost/52

The volume of inventory in a store LocationCube is multiplied by 0.75.The 0.75 is an estimated adjustment since items are always being sold sothat the actual inventory is a bit less than the actual volume ofproduct delivered. This product is divided by the number of items percase. This result is divided by the Sum of the Average Weekly Cases toobtain an average cost per item. Then the Vendors Days Credit is dividedby seven to get Vendor weeks credit, which is subtracted from theaverage cost per item. This difference is multiplied by the product listcost minus any allowances or discounts and back haul discounts anddivided by the number of products per case to obtain the cost perproduct. This product is multiplied by the annual opportunity costdivided by 52 weeks in a year.

10. Location Receiving Stock Cost

The Location Receiving Stock Cost computes the labor cost to stockshelves. The cost depends upon tray and non-tray stocking rates.Products are marked as tray or non-tray stock depending upon theproduct's shelf stocking method.

An algorithm for calculating Location Receiving Stock is as follows:If Product. ShelfStockingMethod=2 ThenLocationRcvgStkCost=((DistributionCenter.StoreCubeNTSecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.StoreCaseNTSecs/Product.CasePack)+DistributionCenter.StorePkgNTSecs)*DistributionCenter.StoreStockLaborRate/3600ElseLocationRcvgStkCost=((DistributionCenter.StoreCubeTraySecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.StoreCaseTraySecs/Product.CasePack)+DistributionCenter.StorePkgTraySecs)*DistributionCenter.StoreStockLabor Rate/3600

In the first branch of the algorithm for a non-tray stocked item, firstthe time in seconds to handle a cubic foot of a non-tray product at astore is multiplied by the cubic feet of each case of the productdivided by the number of products per case to obtain the number ofproducts per second of the non-tray product is stocked on the shelves.The time in unitized seconds to handle a non-tray case of a product at astore divided by the number of products per case yields the time perproduct of handling a product in stocking the shelves. These twoproducts are added to obtain a total number of seconds per product forstocking the non-tray product on the shelves. This total is multipliedby the store stocking labor rate per hour divided by 3600 seconds perhour to obtain the cost of stocking the non-tray product on the shelf.

In the second branch of the algorithm for a tray stocked item, first thetime in seconds to handle a cubic foot of a tray product at a store ismultiplied by the cubic feet of each case of the product divided by thenumber of products per case to obtain the number of products per secondof the tray product is stocked on the shelves. The time in unitizedseconds to handle a tray case of a product at a store divided by thenumber of products per case yields the time per product of handling aproduct in stocking the shelves. These two products are added to obtaina total number of seconds per product for stocking the tray product onthe shelves. This total is multiplied by the stocking labor rate perhour divided by 3600 seconds per hour to obtain the cost of stocking thetray product on the shelf.

11. Product Location Variable Cost Calculation

The ProdLocVariableCost specifier specifies variable cost for eachproduct given a store, demand group, and day. In other embodiments ofthe invention the time period may be a week instead of a day. TheProdLocVariableCost specifier may be computed from the followingequation:ProdLocVariableCost=BagCost+ProductLoclnvCost+CheckoutCostPerPkg+LocationRcvgStkCost+TransCostPerPkg+DCInvCostPerPkg+DCLaborCostPerPkg+HQInvProc

12. Fixed Cost Calculation

ProdLocFixedCost is the Fixed cost for each product given a store,demand group, and day, may be computed from a combination of costscomputed above.

The fixed cost of a product may be calculated by the equation:ProdLocFixedCost=LocationSpaceCost+DCSpaceCostPerStore

13. C_(s,i,k,t) Calculation

Where C_(s,i,k,t), is a cost for a particular product (k) given a store(s), demand group (i), and a day (t), C_(s,i,k,t), may be calculated as:C _(s,i,k,t)=ProdLoeVariableCost+ProdLocFixedCost.

IV. OPTIMIZATION ENGINE AND SUPPORT TOOL

A. Overview

FIG. 4 is a more detailed schematic view of the optimization engine 112and the support tool 116. The optimization engine 112 comprises a ruletool 404 and a price calculator 408. The support tool 116 comprises arule editor 412 and an output display 416.

In operation, the client (stores 124) may access the rule editor 412 ofthe support tool 116 and provides client defined rule parameters (step228). If a client does not set a parameter for a particular rule, adefault value is used. Some of the rule parameters set by the client maybe constraints to the overall weighted price advance or decline,branding price rules, size pricing rules, unit pricing rules, linepricing rules, and cluster pricing rules. These rules will be discussedin more detail regarding the preferred embodiment of the invention. Theclient defined parameters for these rules are provided to the rule tool404 of the optimization engine 112 from the rule editor 412 of thesupport tool 116. Within the rule tool 404, there may be other rules,which are not client defined, such as a group sales equation rule. Therule parameters are outputted from the rule tool 404 to the pricecalculator 408. The demand coefficients 128 and cost data 136 are alsoinputted into the price calculator 408. The client may also provide tothe price calculator 408 through the support tool 116 a desiredoptimization scenario rules. Some examples of scenarios may be tooptimize prices to provide the optimum profit, set one promotional priceand the optimization of all remaining prices to optimize profit, oroptimized prices to provide a specified volume of sales for a designatedproduct and to optimize price. The price calculator 408 then calculatesoptimized prices. The price calculator 408 outputs the optimized pricesto the output display 416 of the support tool 116, which allows thestores 124 to receive the optimized pricing (step 232). In preferredembodiments, the rules and optimizing algorithm are as follows:

B. Preferred Embodiment of Optimization Module

The optimization engine uses the group sales equation and the marketshare equation previously defined in the section on the econometricengine to predict group sales and product market share, respectively.These two are then combined to predict product sales at the store level.The three equations are produced here:

1. First Stage Prediction

The predictive model for the group sales is:

${\ln\left( \frac{{\hat{S}}_{s,i,t}}{S_{s,{B1},t}} \right)} = {{\hat{K}}_{s,i} + {{\hat{\gamma}}_{s,i}\frac{P_{s,i,t}}{{\overset{\_}{P}}_{s,i,t}}} + {{\hat{v}}_{s,1}M_{s,i,t}} + {{\hat{\Psi}}_{s,i}X_{s,i,t}} + {{\hat{\kappa}}_{s,i}X_{s,i,t}\frac{P_{s,i,t}}{{\overset{\_}{P}}_{s,i,t}}} + {\sum\limits_{n = 1}^{\tau}{{\hat{\delta}}_{i,n}\frac{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\; S_{s,i,r}}{\underset{r = {t - {mn}}}{\overset{t - {m{({n - 1})}} - 1}{\sum S_{s,i,r}}}}}} + {\sum\limits_{j \neq i}{{\hat{\phi}}_{s,i,j}\frac{{\hat{S}}_{s,j,t}}{{\overset{\_}{S}}_{s,j,t}}}} + {{\hat{\eta}}_{s,i,t}\left( \frac{{\overset{\_}{P}}_{s,i,t} - {\overset{\overset{\_}{\_}}{P}}_{s,i,t}}{{\overset{\overset{\_}{\_}}{P}}_{s,i,t}} \right)} + {{\hat{\pi}}_{s,i}\frac{{TS}_{s,t}}{T{\overset{\_}{S}}_{s,t}}} + {{\hat{\theta}}_{s,i}\frac{S_{s,i,{t - 7}}}{S_{s,i,{t - 7}}}}}$

2. Second Stage Prediction

The predictive model for estimating the fraction of group sales due to aproduct is:

${\hat{F}}_{s,i,k,t} = \frac{\begin{matrix}{\exp\left\{ {{\hat{\Lambda}}_{s,i,k} + {{\hat{\rho}}_{s,i,k}\left( P_{s,{Ri},k,t} \right)} + {\sum\limits_{p = 1}^{n_{p}}{{\hat{\sigma}}_{s,p,i,k}\left( M_{s,p,i,k,t,} \right)}} +} \right.} \\\left. {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{s,i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\left( F_{s,i,k,r} \right)}}} \right\}\end{matrix}}{\begin{matrix}{\sum\limits_{l \in {Dem}_{i}}{\exp\left\{ {{\hat{\Lambda}}_{s,i,l} + {{\hat{\rho}}_{s,i,l}\left( P_{s,{Ri},l,t} \right)} + {\sum\limits_{p = 1}^{n_{p}}{{\hat{\sigma}}_{s,p,i,l}\left( M_{s,p,i,l,t,} \right)}} +} \right.}} \\\left. {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{s,i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\left( F_{s,i,l,r} \right)}}} \right\}\end{matrix}}$

The predictive model for demand for a given product is then given by{circumflex over (D)}_(s,t,k,t)={circumflex over(F)}_(s,l,k,t)Ŝ_(s,l,t).

3. The Optimization Model

Regression models and the predictive equations derived from the aboveequations are used to construct the optimization model.

The objective is to maximize profit:

${\sum\limits_{i \in G}\;{\sum\limits_{k \in {Dem}_{i}}\;{{\hat{D}}_{s,i,k,t}\left( {P_{s,i,k,t} - C_{s,i,k,t}} \right)}}} = {\sum\limits_{i \in G}{\sum\limits_{k \in {Dem}_{i}}{{\hat{F}}_{s,i,k,t}{S_{s,i,t}\left( {P_{s,i,k,t} - C_{s,i,k,t}} \right)}}}}$and the constraints are:

a. Obey the regression equations governing Ŝ_(l,t)∀i,k

${\ln\left( \frac{{\hat{S}}_{s,i,t}}{S_{s,{Bi},t}} \right)} = {{\hat{K}}_{s,i} + {{\hat{\gamma}}_{s,i}\frac{P_{s,i,t}}{{\overset{\_}{P}}_{s,i,t}}} + {{\hat{v}}_{s,i}M_{s,i,t}} + {{\hat{\Psi}}_{s,i,t}X_{s,i,t}} + {{\hat{\kappa}}_{s,i}X_{s,i,t}\frac{P_{s,i,t}}{{\overset{\_}{P}}_{s,i,t}}} + {\sum\limits_{n = 1}^{\tau}{{\hat{\delta}}_{i,n}\frac{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\; S_{s,i,r}}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\; S_{s,i,r}}}} + {\sum\limits_{j \neq i}{{\hat{\phi}}_{s,i,j}\frac{{\hat{S}}_{s,j,t}}{{\overset{\_}{S}}_{s,j,t}}}} + {{\hat{\eta}}_{s,i,t}\left( \frac{{\overset{\_}{P}}_{s,i,t} - {\overset{\overset{\_}{\_}}{P}}_{s,i,t}}{{\overset{\overset{\_}{\_}}{P}}_{s,i,t}} \right)} + {{\hat{\pi}}_{s,i}\frac{{TS}_{s,t}}{T{\overset{\_}{S}}_{s,t}}} + {{\hat{\theta}}_{s,i}\frac{S_{s,i,{t - 7}}}{S_{s,i,{t - 7}}}}}$

b. Obey the market share equations

${\hat{F}}_{s,i,k,t} = \frac{\begin{matrix}{\exp\;\left\{ {{\hat{\Lambda}}_{s,i,k} + {{\hat{\rho}}_{s,i,k}\left( P_{s,R,i,k,t} \right)} + {\sum\limits_{p = 1}^{n_{p}}{{\hat{\sigma}}_{s,p,i,k}\left( M_{s,p,i,k,t} \right)}} +} \right.} \\\left. {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{s,i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\left( F_{s,i,k,r} \right)}}} \right\}\end{matrix}}{\begin{matrix}{\sum\limits_{1 \in {Dem}_{i}}{\exp\;\left\{ {{\hat{\Lambda}}_{s,i,i} + {{\hat{\rho}}_{s,i,l}\left( P_{s,R,i,l,t} \right)} + {\sum\limits_{p = 1}^{n_{p}}{{\hat{\sigma}}_{s,p,i,l}\left( M_{s,p,i,l,t} \right)}} +} \right.}} \\\left. {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{s,i,k,n}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\left( F_{s,i,l,r} \right)}}} \right\}\end{matrix}}$

C. Constrain price changes to be within a given range of current prices.

PMIN_(s,t,k,t)≦P_(s,i,k,t)≦PMAX_(s,i,k,t)

To simplify notation, the time index may be removed from the equations.Next, all constant terms in each of the expressions may be groupedtogether. So the objective becomes:

Maximize:

$\sum\limits_{i \in G}{\sum\limits_{k \in {Dem}_{i}}{{\hat{F}}_{s,i,k}{S_{s,i}\left( {P_{s,i,k} - C_{s,i,k}} \right)}}}$

The regression equations for Ŝ_(l,t)∀i,k become

$\mspace{20mu}{{\log\left( {\hat{S}}_{s,i} \right)} = {a_{s,i} + {b_{s,i}{\sum\limits_{k \in {Dem}_{i}}{P_{s,i,k}F_{s,i,k}}}} + {\sum\limits_{\underset{j \neq i}{j \in G}}{c_{s,i,j}S_{s,j}}} + {const}_{s,_{i}}}}$where$\mspace{20mu}{{{c_{s}}_{,i,j} = {\frac{{\overset{\_}{S}}_{s,i}}{{\overset{\_}{S}}_{s,j}}{\hat{\phi}}_{s,i,j}}},\mspace{20mu}{b_{s,i} = {\frac{{\overset{\_}{S}}_{s,i}}{{\overset{\_}{P}}_{s,i}}\left( {{\hat{\gamma}}_{s,i} + {\hat{\kappa}X_{s,i}}} \right)}}}$$\mspace{20mu}{a_{s,i} = \frac{{\overset{\_}{S}}_{s,i}\pi_{s,i}}{{\overset{\_}{TS}}_{s}}}$$\mspace{20mu}{{Const}_{s,i} = {{\log\left( {\overset{\_}{S}}_{s,i} \right)} + {\sum\limits_{n = 1}^{\tau}{{\hat{\delta}}_{s,i,n}\frac{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\; S_{s,i,r}}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}\;{\overset{\_}{S}}_{s,i,r}}}} + {{\hat{\Psi}}_{s,i}X_{s,i}} - \gamma_{s,i} + {{\hat{\eta}}_{s,i}\left( {{\overset{\_}{P}}_{s,i,t} - {\overset{\overset{\_}{\_}}{P}}_{s,i,t}} \right)} - {{\hat{\kappa}}_{s,i}X_{s,i}} + {{\hat{v}}_{s,i}R_{s,i}} + {{\hat{\theta}}_{s,i}\frac{S_{s,i,{t - 7}}}{{\overset{\_}{S}}_{s,i,{t - 7}}}} + K_{s,i}}}$Similarly, the regression equations for the market share fractionsbecome:

${\hat{F}}_{s,i,k} = \frac{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k}}{{\overset{\_}{P}}_{s,i,k}}P_{s,i,k}} + {Const}_{s,i,k}} \right\}}{\sum\limits_{l \in {Dem}_{i}}{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{s,i,l}} + {Const}_{s,i,l}} \right\}}}$where

${Const}_{s,i,k} = {{- \rho_{s,i,k}} + {{\hat{\sigma}}_{s,i,l}R_{s,i,k}} + {\sum\limits_{n = 1}^{\tau}{{\hat{\chi}}_{s,i,k,n}\frac{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}F_{s,i,k,r}}{\sum\limits_{r = {t - {mn}}}^{t - {m{({n - 1})}} - 1}{\overset{\_}{F}}_{s,i,k,r}}}} + {\hat{\Lambda}}_{s,i,k}}$With this, the optimization problem may be written as:

4. Problem P1

${Maximize}\text{:}{\sum\limits_{s\; \in {Stores}}\;{\sum\limits_{i \in G}\;{\sum\limits_{k \in {Dem}_{i}}\;{\frac{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k}}{{\overset{\_}{P}}_{s,i,k}}P_{s,i,k}} + {Const}_{s,i,k}} \right\}}{\sum\limits_{l \in {Dem}_{i}}\;{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{s,i,l}} + {Const}_{s,i,l}} \right\}}}{{\hat{S}}_{s,i}\left( {P_{s,i,k} - C_{s,i,k}} \right)}}}}}$Subject to

$\begin{matrix}{a.} & {{{\log\left( {\hat{S}}_{s,i} \right)a_{s,i}} + {b_{s,i}{\sum\limits_{k \in {Dem}_{i}}\;{P_{s,i,k}F_{s,i,k}}}} + {\sum\limits_{\underset{j \neq i}{j \in G}}\;{c_{s,i,j}S_{s,j}}} + {const}_{s,i}},} \\{b.} & {{\hat{F}}_{s,i,k} = {\frac{{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k,}}{{\overset{\_}{P}}_{s,i,k}}P_{s,i,k}} + {Const}_{s,i,k}} \right\}}\;}{\sum\limits_{l \in {Dem}_{i}}\;{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{s,i,l}} + {Const}_{s,i,l}} \right\}}}\mspace{14mu}{and}}} \\{c.} & {{P\;{{MIN}\;}_{s,i,k,}} \leq P_{s,i,k} \leq {P\;{MAX}_{s,i,k}}}\end{matrix}$

Now, in addition to the constraints listed above, the preferredembodiment may model several other business rules via constraints. Theseinclude limits on group price advance or decline, brand pricing rules,size pricing rules, and unit pricing rules.

d. Group price advance or decline.

Since demand groups are made up of like or substitutable products,managers often wish to constrain the overall weighted price advance ordecline in price for them, where the weights are the market shares ofthe products that constitute the demand groups. The constraints are:

${\forall s},i,{{P\;{MIN}_{s,i}} \leq {\sum\limits_{k}\;{F_{s,i,k}P_{s,i,k}}} \leq {P\;{MAX}_{s,i}}}$e. Brand Pricing Rules:

Products are described and categorized by attributes such as brand, sizeand flavor. Therefore, each product may be associated with a set ofattributes and values. These attributes are useful to us for severalreasons. The attributes may be used in the regression modeling in orderto create a parsimonious set of regression coefficients. They may alsobe used in setting rules. For instance, category managers might wish tooffer store brands at lower prices compared to the competing nationalbrands. They might also wish to constrain some product brands to be lessexpensive than others, either for strategic considerations or to meettheir contractual obligations. Specifically, a manager can create a setBrand_(s,i)≡{(p_(s,i,l,),p_(s,i,k) ₂ ):p_(s,i,k) _(l) must cost lessthan p_(s,l,k) ₂ }, which leads to the following constraints: ∀s,i and(p_(s,i,k) ₁ ,p_(s,t,k) ₂ )εBrand_(s,t)p_(s,t,k) ₁ ≦p_(s,t,k) ₂

f. Size Pricing Rules:

Managers might also wish to create rules that relate the price of oneproduct versus another based on their sizes. For instance, they mightwish for products belonging to the same brand and sharing otherattributes, but with different sizes to be priced such that the largersized product costs less per equivalent unit of measure than a smallerone. They cSize_(s,i)≡{(p_(s,i,l) _(t) ,p_(s,i,k) ₂ ):p_(s,i,k) ₁ mustcost less than p_(s,i,k) ₂ an create a set which leads to the followingconstraints:∀s,i and (p _(s,l,k) ₁ ,p _(s,t,k) ₂ )εSize_(s,l) p _(s,t,k) ₂)εSize_(s,l) p _(s,i,k) ₁ ≦p _(s,t,k) ₂

g. Unit Pricing Rules:

Continuing in the same vein, managers might wish to ensure that twoproducts that are identical in every respect but size should be pricedsuch that the larger product costs more than the smaller one. This ruleis closely related to the size pricing rule. To implement this rule,managers can create a set Unit_(s,l)≡{(p_(s,i,l) ₁ ,p_(s,i,k) ₂):e_(s,l,k) ₁ p_(s,i,k) ₁ must cost less than e_(s,l,k) ₂ p_(s,i,k) ₂ },where e_(s,i,k) is the multiplicative factor to convert equivalent unitsinto whole product units. This leads to the following constraints:∀s,i and (p _(s,t,k) ₁ ,p _(s,t,k) ₂ )εUnit_(s,t) e _(s,i,k) ₁ p_(s,i,k) ₁ ≦e _(s,t,k) ₂ p _(s,t,k) ₂

h. Line Pricing Rules:

Retail customers expect common prices for certain groups of productssuch as, for example, cough lozenges of the same brand but differentflavors. These rules may be classified as line pricing rules andimplement them as follows. Product groups called line price groups maybe defined as L₁,l=1, . . . ∥L∥, where every product within a line groupmust have the same price in a store; i.e.,∀L _(l) ,l=1, . . . ∥L∥, and ∀k ₁ ,k ₂ εL _(l) P _(s,i,k) ₁ =P _(s,t,k)₂

i. Cluster Pricing:

Retailers define geographic and other regions within which they maintainthe same price for a given product. This may be translated to the notionof store clusters. A store cluster Cluster_(c) is a set of stores suchthat the price of every product is invariant within the cluster. Inother words, every product has the same price in every store within thecluster although each product may have a different price. In order toimplement these constraints the set of stores may be partitioned intostore clusters Cluster₁, . . . , Cluster_(∥c∥) where C≡{Cluster₁, . . ., Cluster_(∥c∥)}, and define a new price variable P_(c,t,k) whichrepresents the common price for product k in cluster Cluster_(c). Thisvariable may be used in place of the original price variable P_(s,t,k)whenever sεCluster_(c). The entire problem P1 can be rewritten as P2:

5. Problem P2:

Maximize:

$\sum\limits_{c = 1}^{C}\;{\sum\limits_{s \in {Cluster}_{c}}\;{\sum\limits_{i \in G}\;{\sum\limits_{k \in {Dem}_{i}}\;{\frac{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k,}}{{\overset{\_}{P}}_{s,i,k}}P_{c,i,k}} + {Const}_{s,i,k}} \right\}}{\sum\limits_{l \in {Dem}_{i}}\;{\exp\begin{Bmatrix}{{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{c,i,l}} +} \\{Const}_{s,i,l}\end{Bmatrix}}}{{\hat{S}}_{s,i}\left( {P_{c,i,k} - C_{s,i,k}} \right)}}}}}$(Profit Maximizing Objective)Subject to:

$\begin{matrix}{a.} & {{{\log\left( {\hat{S}}_{s,i} \right)a_{s,i}} + {b_{s,i}{\sum\limits_{k \in {Dem}_{i}}\;{P_{s,i,k}F_{s,i,k}}}} + {\sum\limits_{\underset{j \neq i}{j \in G}}\;{c_{s,i,j}S_{s,j}}} +}\mspace{14mu}} \\\; & {{const}_{s,i}\left( {{Group}\mspace{14mu}{Sales}\mspace{14mu}{equation}} \right)} \\{b.} & {{\hat{F}}_{s,i,k} = {\frac{{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k,}}{{\overset{\_}{P}}_{s,i,k}}P_{s,i,k}} + {Const}_{s,i,k}} \right\}}\;}{\sum\limits_{l \in {Dem}_{i}}\;{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{s,i,l}} + {Const}_{s,i,l}} \right\}}}\;\left( {{Market}\mspace{14mu}{Share}\mspace{14mu}{equation}} \right)}} \\{c.} & {{\forall c},{s \in {Cluster}_{c}},i,{{P\;{MIN}_{s,i}} \leq {\sum\limits_{k}\;{F_{s,i,k}P_{s,i,k}}} \leq}} \\\; & {P\;{{MAX}_{s,i}\left( {{Group}\mspace{14mu}{price}\mspace{14mu}{{advance}/{decline}}} \right)}}\end{matrix}$

-   d. PMIN_(ci,k,)≦P_(c,t,k)≦PMAX_(c,t,k), ∀i,k (Product price    advance/decline)-   e. ∀c,i and (p_(c,i,k) ₁ ,p_(c,t,k) ₂ )εBrand_(c,i) p_(c,i,k) ₁    ≦p_(c,l,k) ₂ (Brand Pricing)-   f. ∀c,i and (p_(c,t,k) ₁ ,p_(c,t,k) ₂ )εSize_(c,t) p_(c,t,k) ₁    ≦p_(c,t,k) ₂ (Size Pricing)-   g. ∀c,i, (p_(c,t,k) ₁ ,p_(c,t,k) ₂ )εUnit_(c,i),sεCluster_(c),    e_(s,i,k) ₁ p_(c,i,k) ₁ ≦e_(s,t,k) ₂ p_(c,t,k) ₂ (Unit Pricing)-   h. ∀L_(l),l=1, . . . ∥L∥, ∀k₁,k₂εL_(l),∀c, p_(c,i,k) ₁ =P_(c,t,k) ₂    (Line Pricing)

The optimization problem P2 has the following features: the objectiveand the group sales equations are nonlinear, while the rest of theconstraints are linear. In the preferred embodiment, a heuristicapproach is used to generate a good feasible starting point for theproblem, if one exists. The heuristic works by repeatedly solving arelated problem involving all the linear constraints 4-8, a set oflinear constraints that approximate the group price advance and declineconstraints (3), and an objective that involves the marginalcontribution to profit of each product.

6. Heuristic to Generate the Feasible Solution:

The problem may be decomposed by store cluster and hence attention maybe restricted to a single cluster Cluster_(s). for this explanation.

In the preferred embodiment, the heuristic focuses on finding goodinitial feasible solutions to the problem. First, a related problem isdefined.

The market shares of all products are calculated based on a set ofinitial prices. Then these initial market shares are used as fixedweights to develop linear constraints that limit the advance and declineof demand group prices at each store. Specifically, let

${{\overset{\_}{p}}_{{Cluster}_{c},i} = \begin{pmatrix}P_{c,i,k} \\\vdots \\P_{c,i,{{Dem}_{i}}}\end{pmatrix}},{{\overset{\_}{p}}_{{Cluster}_{c} =}\begin{pmatrix}{\overset{\_}{p}}_{{Cluster}_{c},1} \\\vdots \\{\overset{\_}{p}}_{{Cluster}_{c},{G}}\end{pmatrix}},{and}$${{G_{s,i,k}\left( {\overset{\_}{p}}_{{Cluster}_{c},i} \right)} \equiv {\frac{{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k,}}{{\overset{\_}{P}}_{s,i,k}}P_{c,i,k}} + {Const}_{s,i,k}} \right\}}\;}{\sum\limits_{l \in {Dem}_{i}}\;{\exp\begin{Bmatrix}{{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{c,i,l}} +} \\{Const}_{s,i,l}\end{Bmatrix}}}{\forall{s \in {Cluster}_{c}}}}},$the market share of product k at store s given price vector p _(Cluster)_(c) _(,t) at the store for demand group i. Then, if the initial pricevectors are q _(Cluster) _(c) _(,t), ∀sεCluster_(c),i, the approximategroup price constraints are defined to be:

${\forall{s \in {Cluster}_{c}}},{\forall i},{{PMIN}_{s,i} \leq {\sum\limits_{k}\;{{G_{s,i,k}\left( {\overset{\_}{q}}_{{Cluster}_{c},i} \right)}P_{c,i,k}}} \leq {{PMAX}_{s,i}.}}$

Note that the weights G_(s,l,k)( q _(Cluster) _(c) _(,t)) are constantsand so the constraints are linear in P_(c,t,k). Now problem P3 isdefined as follows:

7. Problem P3( q _(Cluster) _(c) )

${Maximize}\text{:}{\sum\limits_{s\; \in {Cluster}_{c}}\;{\sum\limits_{i \in G}\;{\sum\limits_{k \in {Dem}_{i}}\;{\frac{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,k}}{{\overset{\_}{P}}_{s,i,k}}P_{c,i,k}} + {Const}_{s,i,k}} \right\}}{\sum\limits_{l \in {Dem}_{i}}\;{\exp\left\{ {{\frac{{\hat{\rho}}_{s,i,l}}{{\overset{\_}{P}}_{s,i,l}}P_{c,i,l}} + {Const}_{s,i,l}} \right\}}}\left( {P_{c,i,k} - C_{s,i,k}} \right)}}}}$

Subject to:

${\forall_{s}{\in {Cluster}_{c}}},{\forall_{i}{,{{P\;{MIN}_{s,i}} \leq {\sum\limits_{k}{{G_{s,i,k}\left( {\overset{\_}{q}}_{{Cluster}_{c},i} \right)}P_{c,i,k}}} \leq {P\;{MAX}_{s,i}}}}}$(Group Pricing-approximate)

-   2. PMIN_(c,t,k)≦P_(c,t,k)≦PMAX_(c,t,k), ∀i,k (Product price    advance/decline)-   3. ∀i and (p_(c,t,k) ₁ ,p_(c,t,k) ₂ )εBrand_(c,i)p_(c,i,k) ₁    ≦p_(c,l,k) ₂ (Brand Pricing)-   4. ∀i and (p_(c,t,k) ₁ ,p_(c,i,k) ₂ )≦p_(c,i,k) ₂ (Size Pricing)-   5. ∀i, (p_(c,t,k) ₁ ,p_(c,t,k) ₂ )εUnit_(c,t)sεCluster_(c),    e_(s,i,k) ₁ p_(c,t,k) ₁ ≦e,s,t,k ₁ p_(c,i,k) ₂ (Unit Pricing)-   6. ∀L_(l),l=1, . . . ∥L∥, ∀k₁,k₂εL_(l),P_(c,t,k) ₁ =P_(c,t,k) ₂    (Line Pricing)

Problem P3 has a nonlinear objective but only linear constraints. Notethat the Problem P3 does not include group sales. The objectivemaximizes the sum of marginal profit for all products over all thestores in the cluster. This approach may be easily extended to include amultiplier that approximates the group sales of all groups over allstores.

To facilitate understanding, FIG. 6 is a flow chart of the preferredembodiment of the invention for providing a good initial feasiblesolution derived by solving Problem P3 iteratively as follows:

-   1. Set current incumbent prices to equal initial prices, which are    prices currently used by the stores (step 604).-   2. Set q_(cluster) _(c) to be the set current incumbent prices for    products at the stores in the cluster. Calculate the product market    shares for each product at each store and generate the approximate    group price constraints as a function of q_(Cluster) _(c) (Step    606). Solve problem P3( q _(Cluster) _(c) ) for the optimal solution    P _(Cluster) _(c) * (Step 608). Check to see if problem is feasible    (Step 612). If the problem is infeasible, no feasible solution    exists, terminate optimization (Step 616).-   3. Compare the optimal solution P _(Cluster) _(c) * to the problem    P3( q _(Cluster) _(c) ) to q _(Cluster) _(c) (Step 620). If they are    the same, then go to step 5. Else-   4. Let q _(Cluster) _(c) = P _(Cluster) _(c) * (Step 624) and go to    step 2.-   5. Redefine the bounds PMIN_(c,I) and PMAX_(c,I) as follows:

${P\;{MIN}_{s,i}} \equiv {{MIN}\left( {{P\;{MIN}_{s,i}},{\sum\limits_{k}{{G_{s,i,k}\left( {\overset{\_}{p}}_{{Cluster}_{c},i} \right)}P_{c,i,k}}}} \right)}$and

${P\;{MAX}_{s,i}} \equiv {{{MAX}\left( {{P\;{MAX}_{s,i}},{\sum\limits_{k}{{G_{s,i,k}\left( {\overset{\_}{p}}_{{Cluster}_{c},i} \right)}P_{c,i,k}}}} \right)}.}$This ensures that P _(Cluster) _(c) * is feasible for P2 (Step 628).

-   6. Compute the group sales corresponding to P _(Cluster) _(c) *.    This is done by solving P2 with p _(Cluster) _(c) fixed (Step 632).-   7. Use resulting optimal price vector P _(Cluster) _(c) * and group    sales as a feasible starting solution to P3 (Step 636).

In practice, convergence is declared in step 2 when the change in pricesis below an acceptable tolerance threshold.

This method turns out to have several advantages. First, it finds aninitial feasible solution by solving iteratively a nonlinear problemwith linear constraints. Problem P3 is considerably easier to solve thanProblem P1 and Problem P2. Second, the initial feasible solution turnsout to be a good starting solution to Problem P2. It converges quitequickly from this starting point. Therefore these initial feasiblesolutions are provided as a starting point for solving Problem P2.

An example of this would have two products A and B. Each would becurrently priced in the store at $1. In this example the rules onlyprovide one constraint, which in this example is the Group priceadvance/decline constraint so that

${\forall_{c,s}{\in {Cluster}_{c}}},i,{{P\;{MIN}_{s,i}} \leq {\sum\limits_{k}{F_{s,i,k}P_{s,i,k}}} \leq {P\;{{MAX}_{s.i}.}}}$Therefore, for step 604, the current incumbent prices are set equal tothe current price in the store ($1). Using the demand and market sharemodels, the market share for each product A and B is calculated for thegiven incumbent prices of $1. For example, the market share of A may becalculated to be 50% (0.5) and the market share of B may be calculatedto be 50% (0.5). Therefore, the constraint

${P\;{MIN}_{s,i}} \leq {\sum\limits_{k}{F_{s,i,k}P_{s,i,k}}} \leq {P\;{MAX}_{s,i}}$may be approximated by PMIN_(s,i)≦F_(A)P_(A)+F_(B)P_(B)≦PMAX_(s,i),which yields the constraint equationPMIN_(s,i)≦(0.5)P_(A)+(0.5)P_(B)≦PMAX_(s,l) (Step 606). The presentincumbent prices of $1, the linear constraint equation, and thenon-linear objective of maximizing profit are solved with a non-linearoptimization problem software package to find optimized prices when theconstraint equation is PMIN_(s,i)≦(0.5)P_(A)+(0.5)P_(B)≦PMAX_(s,i) (Step608). An example of such non-linear optimization problem softwarepackages are MINOS™, which was invented by Bruce Murtagh and MichaelSaunders and is marketed by Stanford Business Software, Inc, andCONOPT™, which was written by Arne Stolbjerg Drud and marketed by ARKIConsulting and Development A/S. Such software packages would provide aflag if the problem is not feasible (Steps 612 and 616).

In this example the non-linear optimization software package provides anoptimized price for product A to be $0.70 and an optimized price forproduct B to be $1.20. The current incumbent prices are compared to theoptimized price. Since for product A $1≠$0.70 or for product B $1≠$1.20(Step 620), then the incumbent prices are set equal to the optimizedprice (Step 624). From the new incumbent prices new market shares arecalculated, using the model for predicting product share. For example,the new calculated market share may be 65% (0.65) for Product A and 35%(0.35) for Product B. The new market shares are placed into theconstraint equation to provide a new constraint equation (Step 606),such as PMIN_(s,i)≦(0.65)P_(A)+(0.35)P_(B)≦PMAX_(s,i). The presentincumbent prices of $0.70 and $1.20, the linear constraint equation, andthe non-linear objective of maximizing profit are solved with anon-linear optimization problem software package to find optimizedprices when the constraint equation isPMIN_(s,i)≦(0.65)P_(A)+(0.35)P_(B)≦PMAX_(s,i) (Step 608).

In this example the non-linear optimization software package provides anoptimized price for Product A to be $0.70 and an optimized price forProduct B to be $1.20. Since the optimized prices are now equal to (orare within a threshold delta) the incumbent prices (Step 620), the groupprice bonds are reset so that the optimized prices for Product A ($0.70)and Product B ($1.20) are feasible for Problem P2 (Step 628). Theoptimized prices are placed in the demand equation to generate groupsales (Step 632). The optimized prices and group sales are then used asa feasible starting solution to Problem P3 (Step 636). This means thatthe optimized prices, or Product A ($0.70) and Product B ($1.20), thegenerated group sales, the Profit Optimization Object, and theconstraints of Problem P2 are provided to a non-linear optimizationproblem software, such as MINOS or CONOPT. The use of the optimizedprices allows for a quick convergence and an initial feasible solutionfor the solving of Problem P2. The initial feasible solution isimportant because in a non-linear equation there may be several localoptimal locations and there are several areas where a solution is notfeasible or there is no convergence or convergence is very slow. Theinitial feasible solution provides a feasible local optimal locationwhere there is a faster convergence.

In general, optimizations with a linear objective and linear constraintsare the easiest to solve. Optimizations with non-linear objectives andlinear constraints may be harder to solve. Optimizations with non-linearobjectives with non-linear constraints may be the hardest to solve.Therefore the heuristic uses an iterative approximation to convert anoptimization with a non-linear objective and non-linear constraints toan optimization with a non-linear objective and linear constraints toprovide an initial solution. Once the initial solution is solved, theinitial solution is used as a starting point to optimize a non-linearobjective with non-linear constraints. Such optimizations may provide agradient around the initial solution to determine a direction in whichto go. Prices are used in that direction to obtain a next point. Agradient around the new prices are then used to determine a newdirection. This may be iteratively done until a boundary is reached oruntil the prices do not change. Other optimization heuristics and otherapproaches may be used in different embodiments of the invention.

The use of calculating demand group demand as a function of price andthen using market share to calculate a product's demand from the demandgroup demand in an optimization scheme is a novel aspect of theinvention.

FIGS. 7A and 7B illustrate a computer system 900, which forms part ofthe network 10 and is suitable for implementing embodiments of thepresent invention. FIG. 7A shows one possible physical form of thecomputer system. Of course, the computer system may have many physicalforms ranging from an integrated circuit, a printed circuit board, and asmall handheld device up to a huge super computer. Computer system 900includes a monitor 902, a display 904, a housing 906, a disk drive 908,a keyboard 910, and a mouse 912. Disk 914 is a computer-readable mediumused to transfer data to and from computer system 900.

FIG. 7B is an example of a block diagram for computer system 900.Attached to system bus 920 are a wide variety of subsystems.Processor(s) 922 (also referred to as central processing units, or CPUs)are coupled to storage devices, including memory 924. Memory 924includes random access memory (RAM) and read-only memory (ROM). As iswell known in the art, ROM acts to transfer data and instructionsuni-directionally to the CPU and RAM is used typically to transfer dataand instructions in a bi-directional manner. Both of these types ofmemories may include any suitable of the computer-readable mediadescribed below. A fixed disk 926 is also coupled bi-directionally toCPU 922; it provides additional data storage capacity and may alsoinclude any of the computer-readable media described below. Fixed disk926 may be used to store programs, data, and the like and is typically asecondary storage medium (such as a hard disk) that is slower thanprimary storage. It will be appreciated that the information retainedwithin fixed disk 926 may, in appropriate cases, be incorporated instandard fashion as virtual memory in memory 924. Removable disk 914 maytake the form of any of the computer-readable media described below.

CPU 922 is also coupled to a variety of input/output devices, such asdisplay 904, keyboard 910, mouse 912 and speakers 930. In general, aninput/output device may be any of: video displays, track balls, mice,keyboards, microphones, touch-sensitive displays, transducer cardreaders, magnetic or paper tape readers, tablets, styluses, voice orhandwriting recognizers, biometrics readers, or other computers. CPU 922optionally may be coupled to another computer or telecommunicationsnetwork using network interface 940. With such a network interface, itis contemplated that the CPU might receive information from the network,or might output information to the network in the course of performingthe above-described method steps. Furthermore, method embodiments of thepresent invention may execute solely upon CPU 922 or may execute over anetwork such as the Internet in conjunction with a remote CPU thatshares a portion of the processing.

In addition, embodiments of the present invention further relate tocomputer storage products with a computer-readable medium that havecomputer code thereon for performing various computer-implementedoperations. The media and computer code may be those specially designedand constructed for the purposes of the present invention, or they maybe of the kind well known and available to those having skill in thecomputer software arts. Examples of computer-readable media include, butare not limited to: magnetic media such as hard disks, floppy disks, andmagnetic tape; optical media such as CD-ROMs and holographic devices;magneto-optical media such as floptical disks; and hardware devices thatare specially configured to store and execute program code, such asapplication-specific integrated circuits (ASICs), programmable logicdevices (PLDs) and ROM and RAM devices. Examples of computer codeinclude machine code, such as produced by a compiler, and filescontaining higher level code that are executed by a computer using aninterpreter.

FIG. 8 is a schematic illustration of an embodiment of the inventionthat functions over a computer network 800. The network 800 may be alocal area network (LAN) or a wide area network (WAN). An example of aLAN is a private network used by a mid-sized company with a buildingcomplex. Publicly accessible WANs include the Internet, cellulartelephone network, satellite systems and plain-old-telephone systems(POTS). Examples of private WANs include those used by multi-nationalcorporations for their internal information system needs. The network800 may also be a combination of private and/or public LANs and/or WANs.In such an embodiment the price optimizing system 100 is connected tothe network 800. The stores 124 are also connected to the network 800.The stores 124 are able to bi-directionally communicate with the priceoptimizing system 100 over the network 800.

V. EXAMPLES

In an example of the invention, using a single category for a chain ofstores, the category had 430 products, 6 demand groups, and 433 stores.Table 1 shows the number of products in each demand group. Within eachdemand group products compete against each other, such as an 8 oz. Cokeand a 24 oz. Coke. Within the category products may compete or may becomplementary. An example of a category may be drinks. In this example,12 months of scanner data was collected from the stores to provide theeconometric demand model coefficients. In this example costs were notdirectly accounted for, but instead profit was measured by thedifference between selling price and the unit cost.

TABLE 1 Demand Group A B C D E F Number of Products 5 44 18 17 74 21

The econometric demand model coefficients were calculated by the use ofthe SAS™ program provided by SAS Institute Inc., of Cary N.C. An erroranalysis was performed on the demand model to determine a confidenceinterval for one standard deviation (68%). Table 2 shows the error for aone standard deviation confidence interval. Pct. Error is measured as100*(forecasted value−actual value)/(forecasted value), and Abs. Pct.Error is calculated as the absolute value of Pct. Error.

TABLE 2 Error Error Weighted Weighted ErrorWeighted Average StandardAverage Level N Type Mean Deviation Median Store Product 22821 Pct.Error(Volume) −13.22% 46.20% −1.97% Store Product 22821 Abs.Pct.Error(Volume) 47.54% 32.61% 38.48% Store Product 22821 Pct.Error (Profit)−13.26% 46.59% −2.01% Store Product 22821 Abs.Pct.Error (Profit) 47.62%32.97% 38.52% Store Demand 2361 Pct.Error (Volume) −0.45% 14.11% −1.06%Group Store Demand 2361 Abs.Pct.Error (Volume) 12.87% 9.34% 12.12% GroupStore Demand 2361 Pct.Error (Profit) −0.57% 17.44% 0.52% Group StoreDemand 2361 Abs.Pct.Error (Profit) 14.34% 12.28% 12.99% Group StoreCategory 394 Pct.Error (Volume) 0.15% 7.27% 0.32% Store Category 394Abs.Pct.Error (Volume) 6.27% 4.77% 5.86% Store Category 394 Pct.Error(Profit) 0.87% 7.59% 1.52% Store Category 394 Abs.Pct.Error (Profit)6.80% 5.02% 6.41%

Table 2 shows that at the Store Category level, the error weightedstandard deviation in volume of sales between predicted sales and actualsales was 7.27%. This means that there is a 68% confidence (one standarddeviation) that the actual sales would fall within 7.27% of predictedsales or in other words the probability that the actual price would bemore than one standard deviation away from the predicted price would be32%. Examining the Store Demand Group level the error weighted standarddeviation in volume increases to 14.11%. At the product level, the errorweighted standard deviation in volume increases to 46.20%. Generally,the stores in this example carry extra inventory as safety stock, maybeas long as 4 weeks. It has been estimated that the 46.20% error would becovered by the general safety stock policies in most stores.

In this example group prices were allowed to vary between 85% and 105%of their current prices and the product prices were allowed to varybetween 70% and 115% of their current levels. Therefore in this examplePMIN_(s,i)

there were only two constraints, which were:

-   1.

${\forall_{s}{\in {Cluster}_{c}}},{\forall_{i}{,{{P\;{MIN}_{s,i}} \leq {\sum\limits_{k}{{G_{s,i,k}\left( {\overset{\_}{q}}_{{Cluster}_{c},i} \right)}P_{c,i,k}}} \leq {p\;{MAX}_{s,i}}}}}$(Group Pricing—approximate) with PMIN being 85% of current prices andPMAX being 105% of current prices.

-   2. PMIN_(ci,k)≦P_(c,t,k)≦PMAX_(c,t,k)∀i,k (Product price    advance/decline) with PMIN being 70% of current prices and PMAX    being 115% of current prices

Profit was then optimized to provide new optimized prices, using the twoconstraints. This optimization is of profit is measured at the categorylevel, since aggregated profit is of more interest than profit by demandgroup. This is because a store may be willing to have lower profits fora single demand group if the profit of an entire category is raised.

FIG. 9A is a graph of original profit 964 from actual sales of the storeusing actual prices and optimal profit 968 from optimized salesresulting from the calculated optimized prices bounded by itsprobability. Along the x axis is profit in dollars. Along the y axis isthe probability of achieving a particular profit value. The probabilityof getting at least a certain dollar profit is equal to the area underthe right side of the curve. For the original profit there was a highestprobability of obtaining between $1,200 and $1,600 profit. For theoptimal profit there was a highest probability of obtaining between$3,200 and $4,800 profit. Thus showing a possible two fold increase inprofit.

FIG. 9B is a graph 972 of percentage increase in profit calculated bythe optimal profit minus the original profit the difference divided bythe original profit, Δ%=(P_(optimal)−P_(original))/P_(original), and theprobability of obtaining at least that percentage increase in profit.The x axis is the percentage increase in profit and the y axis is theprobability of at least obtaining that percentage increase. According tothis graph there is almost a 100% chance of increasing profit at least104.3%.

In the specification, examples of product are not intended to limitproducts covered by the claims. Products may for example include food,hardware, software, real estate, financial devices, intellectualproperty, raw material, and services. The products may be sold wholesaleor retail, in a brick and mortar store or over the Internet, or throughother sales methods.

While this invention has been described in terms of several preferredembodiments, there are alterations, permutations, and substituteequivalents, which fall within the scope of this invention. It shouldalso be noted that there are many alternative ways of implementing themethods and apparatuses of the present invention. It is thereforeintended that the following appended claims be interpreted as includingall such alterations, permutations, and substitute equivalents as fallwithin the true spirit and scope of the present invention.

1. A computer-implemented method for determining a preferred set ofprices for a plurality of products, the method being implemented as aplurality of program instructions in a computer system, the methodcomprising: receiving, using the computer system, a plurality of demandcoefficients; receiving, using the computer system, known cost dataincluding activity-based costs; inputting, using the computer system,missing or incomplete cost data to give inputted cost data; combiningusing the computer system, said known cost data with said inputted costdata to give a combined cost data set; generating, using the computersystem, a sales model from the plurality of demand coefficients;generating, using the computer system, a cost model from the combinedcost data set, and wherein the activity-based costs include variablecosts and fixed costs, further wherein said cost model determines atotal cost for each product in a given demand group in a given store fora given time period by computing a cost for each selected costingactivity; receiving, using the computer system, a set of actual prices;initializing, using the computer system, a set of incumbent prices tothe set of actual prices; generating a local optimum by applying, usingthe computer system, the set of incumbent prices to the sales model andthe cost model wherein the local optimum for the preferred set of pricesmaximizes profit; and generating the preferred set of prices byapplying, using the computer system, the local optimum prices to thesales model and the cost model in an iterative manner until thepreferred set of prices is reached.
 2. The method, as recited in claim1, further comprising specifying, using the computer system, a pluralityof rules, and wherein the generation of the preferred set of pricescomprises: determining, using the computer system, a set of startingprices; and wherein the generation of the local optimum for thepreferred set of prices for the plurality of products includes complyingwith the plurality of rules, further wherein said plurality of rulesconstrain the preferred set of prices to fall within limits conformingto business strategy.
 3. The method, as recited in claim 1, wherein thecost data is from an individual store and the preferred set of prices isgenerated for said individual store.
 4. The method, as recited in claim1, wherein the cost data is from a cluster of stores and the preferredset of prices is generated for said cluster of stores.
 5. The method, asrecited in claim 1, further comprising: generating, using the computersystem, equivalent prices for said plurality of products by dividingindividual product prices by a standardized unit of measure; andincorporating, using the computer system, said equivalent prices intosaid sales model.
 6. The method, as recited in claim 1, wherein saiddemand group is a set of highly substitutable products.
 7. An apparatusfor determining a preferred set of prices for a plurality of products,the apparatus comprising: an econometric engine configured to receive aplurality of demand coefficients and to generate a sales model from theplurality of demand coefficients; a financial engine configured toreceive known cost data including activity-based costs and to imputemissing or incomplete cost data to give imputed cost data, the financialengine also configured to combine said known cost data with said imputedcost data to give a combined cost data set, the financial engine furtherconfigure to generate a cost model from the combined cost data set, andwherein the activity-based costs include variable costs and fixed costs,further wherein said cost model determines a total cost for each productin a given demand group in a given store for a given time period bycomputing a cost for each selected costing activity; and an optimizationengine configured to receive a set of actual prices and to initialize aset of incumbent prices to the set of actual prices, the optimizationengine also configured to generate a local optimum by applying the setof incumbent prices to the sales model and the cost model wherein thelocal optimum for the preferred set of prices maximizes profit, theoptimization engine further configured to generate the preferred set ofprices by applying the local optimum prices to the sales model and thecost model in an iterative manner until the preferred set of prices isreached.
 8. The apparatus as recited in claim 7, wherein theoptimization engine is further configured to specify a plurality ofrules, and wherein the generation of the preferred set of pricesincludes determining a set of starting prices; and wherein thegeneration of the local optimum for the preferred set of prices for theplurality of products includes complying with the plurality of rules,further wherein said plurality of rules constrain the preferred set ofprices to fall within limits conforming to business strategy.
 9. Theapparatus as recited in claim 7, wherein the cost data is from anindividual store and the preferred set of prices is generated for saidindividual store.
 10. The apparatus as recited in claim 7, wherein thecost data is from a cluster of stores and the preferred set of prices isgenerated for said cluster of stores.
 11. The apparatus as recited inclaim 7, wherein the econometric engine is further configured togenerate equivalent prices for said plurality of products by dividingindividual product prices by a standardized unit of measure, and toincorporate said equivalent prices into said sales model.
 12. Theapparatus as recited in claim 7, wherein said demand group is a set ofhighly substitutable products.